Long-Baseline Laser Interferometry for the Detection of Binary Black-Hole Mergers Citation Hall, Evan Drew (2017) Long-Baseline Laser Interferometry for the Detection of Binary Black-Hole Mergers. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9PG1PQ9. https://resolver.caltech.edu/CaltechTHESIS:01302017-113507797 Abstract Late in 2015, gravitational physics reached a watershed moment with the first direct detections of gravitational waves. Two events, each from the coalescence of a binary black hole system, were detected by the Laser Interferometer Gravitational-wave Observatory (LIGO). At present, LIGO comprises two 4 km laser interferometers, one in Washington and the other in Louisiana; a third detector is planned to be installed in India. These interferometers, known as Advanced LIGO, belong to the so-called “second generation” of gravitational-wave detectors. Compared to the first-generation LIGO detectors (Initial and Enhanced LIGO), these instruments use multi-stage active seismic isolation, heavier and higher-quality mirrors, and more laser power to achieve an unprecedented sensitivity to gravitational waves. In 2015, both Advanced LIGO detectors achieved a strain sensitivity better than 10 -23 /Hz 1/2 at a few hundred hertz; ultimately, these detectors are designed to achieve a sensitivity of a few parts in 10 -24 /Hz 1/2 at a few hundred hertz. This thesis covers several topics in gravitational physics and laser interferometry. First, it presents the design, control scheme, and noise performance of the Advanced LIGO detector in Washington during the first observing run (O1). Second, it discusses some issues relating to interferometer calibration, and the impact of calibration errors on astrophysical parameter estimation. Third, it discusses the prospects for using terrestrial and space-based laser interferometers as dark matter detectors. This thesis has the internal LIGO document number P1600295. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: gravitational waves; laser interferometers Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Adhikari, Rana Group: LIGO Thesis Committee: Adhikari, Rana (chair) Chen, Yanbei Golwala, Sunil Weinstein, Alan Jay Defense Date: 14 November 2016 Other Numbering System: Other Numbering System Name Other Numbering System ID LIGO Document Control Center P1600295 Funders: Funding Agency Grant Number National Science Foundation PHY-0757058 Record Number: CaltechTHESIS:01302017-113507797 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:01302017-113507797 DOI: 10.7907/Z9PG1PQ9 Related URLs: URL URL Type Description https://dcc.ligo.org/LIGO-P1600295 Organization LIGO Document Control Center (password required for access) https://arxiv.org/abs/1605.01103 arXiv Article adapted for ch. 6 https://clrccires.colorado.edu/4pagesummaries/W/W2.pdf Organization Figure adapted for ch. 2 http://dx.doi.org/10.1103/PhysRevD.93.112004 DOI Figure adapted for ch. 3 ORCID: Author ORCID Hall, Evan Drew 0000-0001-9018-666X Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10031 Collection: CaltechTHESIS Deposited By: Evan Hall Deposited On: 13 Feb 2017 20:11 Last Modified: 26 Oct 2021 18:19 Thesis Files Preview PDF - Final Version See Usage Policy. 14MB Repository Staff Only: item control page

Long-baseline laser interferometry for the detection of binary black-hole mergers Thesis by Evan D. Hall In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2017 (Defended 14 November 2016) ii © 2017 Evan D. Hall ORCID: 0000–0001–9018–666X All rights reserved except where otherwise noted iii Abstract Late in 2015, gravitational physics reached a watershed moment with the first direct detections of gravitational waves. Two events, each from the coalescence of a binary black hole system, were detected by the Laser Interferometer Gravitationalwave Observatory (LIGO). At present, LIGO comprises two 4 km laser interferometers, one in Washington and the other in Louisiana; a third detector is planned to be installed in India. These interferometers, known as Advanced LIGO, belong to the so-called “second generation” of gravitational-wave detectors. Compared to the first-generation LIGO detectors (Initial and Enhanced LIGO), these instruments use multi-stage active seismic isolation, heavier and higher-quality mirrors, and more laser power to achieve an unprecedented sensitivity to gravitational waves. In 2015, both Advanced LIGO detectors achieved a strain sensitivity better than 10−23 /Hz1/2 at a few hundred hertz; ultimately, these detectors are designed to achieve a sensitivity of a few parts in 10−24 /Hz1/2 at a few hundred hertz. This thesis covers several topics in gravitational physics and laser interferometry. First, it presents the design, control scheme, and noise performance of the Advanced LIGO detector in Washington during the first observing run (O1). Second, it discusses some issues relating to interferometer calibration, and the impact of calibration errors on astrophysical parameter estimation. Third, it discusses the prospects for using terrestrial and space-based laser interferometers as dark matter detectors. This thesis has the internal LIGO document number P1600295. iv Acknowledgments Over the past years, I have benefited from the support of many wise and wonderful people without whom this work would not have been possible. Thanks to Steve and Sam for throwing me into the deep end with electronics and lasers at Chicago. And thanks to Bobby, Lee, Brittany, and everyone else in the old meson tunnel for putting up with me. Thanks to Rana for the opportunity to work on optical cavities from 4 cm to 4 km, for always being a source of advice and motivation, and for going to bat for his students when needed. Thanks to Nic and Jamie for dealing with my incessant questions. Thanks to Tara, David, Zach, Eric, Koji, Aidan, and Gabriele for making the subbasement a little less monastic. Thanks to Daniel, Keita, and Peter for always being able to dredge up a plan of attack from the bog of loop tweaks, signal rewiring, and noise projections at the Hanford detector. Thanks to Sheila, Kiwamu, Jenne, Jeff, Alexa, Dan, Terra, and everyone else at Hanford who was part of the mad dash to the first observing run. And thanks to Stefan, Matt, and Lisa for showing up from time to time to help us not repeat the mistakes of the past. Finally, I’d like to thank my parents and family for their many years of unwavering support. v ̇̇ ̇‫ו̇תם מן לא ינתהי לדר̇ ̇̇גה י̇צי ̇טלאמה בברק בל ב̇גסם סקיל או נחוה מן ﭏח‬ ‫גארה‬ ‫וגירהא ﭏתי ת̇צי פי ̇טלמאת ﭏליל ולו ̇דלך ﭏ̇צו ﭏיסיר אי̇צא ﭏ̇די ישרק עלינא‬ ‫ליס הו דאימא בל ילוח וי̇כפי כאנה להט החרב המתהפכת‬ ‫— רמב״ם‬ And then there are some whose darkness is illumined not by lightning, but by some kind of crystal or similar stone, or other substance that possesses the property of shining during the night; and to them even this small amount of light is not continuous, but now it shines and now it vanishes, as if it were the Flame of the Rotating Sword. — Maimonides vi TABLE OF CONTENTS Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . xii Chapter I: General relativity and gravitational radiation . . . . . . . . . . . . . 1 1.1 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Weak-field limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Gravitational radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Terrestrial detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Other detection methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter II: The Advanced LIGO interferometers: design and control . . . . . . 8 2.1 Topology overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Laser frequency control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Differential arm length control . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4 Noise from vertex length controls . . . . . . . . . . . . . . . . . . . . . . 32 2.5 Angular control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.6 Suspension control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.7 Intensity control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Chapter III: Noise in the Advanced LIGO detectors . . . . . . . . . . . . . . . . . 55 3.1 Quantum noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Seismic and Newtonian noise . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.4 Gas noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5 Frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6 Intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.7 Photodiode noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.8 Actuator noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.9 Noise from vertex degrees of freedom . . . . . . . . . . . . . . . . . . . . 79 3.10 Noise from angular degrees of freedom . . . . . . . . . . . . . . . . . . . 80 3.11 Jitter noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.12 Noise budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Chapter IV: Observational results . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.1 GW150914 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 LVT151012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.3 GW151226 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter V: Some topics in calibration . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 vii 5.2 Differential arm length optomechanical plant . . . . . . . . . . . . . . 89 5.3 Systematic calibration errors . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Optimal transfer function measurements from the Fisher matrix . . 99 Chapter VI: Searching for dark matter with laser interferometers . . . . . . . . 111 6.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 Newtonian and Yukawa-type interactions . . . . . . . . . . . . . . . . . 112 6.3 Detection with laser interferometers . . . . . . . . . . . . . . . . . . . . 113 6.4 Search strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Chapter VII: Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1 Interferometery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Appendix A: Preliminary notions about loops . . . . . . . . . . . . . . . . . . . . 125 A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 A.2 Crossovers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A.4 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Appendix B: Median estimation of amplitude spectral densities . . . . . . . . . 131 B.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 B.2 Gauss and Rayleigh distributions . . . . . . . . . . . . . . . . . . . . . . 132 B.3 rms versus median averaging . . . . . . . . . . . . . . . . . . . . . . . . 133 B.4 Real-world example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Appendix C: Optical cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 C.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 C.2 Pound–Drever–Hall locking . . . . . . . . . . . . . . . . . . . . . . . . . . 139 C.3 Michelson locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Appendix D: Some algebra for the differential arm length plant . . . . . . . . . 143 Appendix E: In-situ test mass ringdown measurement . . . . . . . . . . . . . . . 146 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 viii LIST OF ILLUSTRATIONS Number Page 1.1 Strain sensitivities for some past, current, and future gravitational wave experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Diagram of a Michelson interferometer. . . . . . . . . . . . . . . . . . . . 9 2.2 Diagram of a Fabry–Pérot interferometer. . . . . . . . . . . . . . . . . . . 11 2.3 Overview of the Advanced LIGO interferometer. . . . . . . . . . . . . . . 16 2.4 Diagram of the Advanced LIGO frequency and intensity stabilization systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Open-loop transfer function of the reference cavity stabilization servo 2.6 Open-loop transfer function (model and measurement) of the input 20 modecleaner frequency locking loop, with no feedback from the interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.7 Open-loop transfer function (model and measurement) of the laser frequency locking to the interferometer common-mode arm length. . . 24 2.8 Crossover (model and measurement) from the laser frequency feedback to the modecleaner length feedback. . . . . . . . . . . . . . . . . . . 25 2.9 Block diagram of the Advanced LIGO frequency stabilization system. . 26 2.10 Differential arm length optomechanical response as a function of SRC detuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.11 Open-loop transfer function of the differential arm length control. . . . 31 2.12 Crossover transfer functions for the penultimate and upper-intermediate actuator offloading for the differential arm length control. . . . . . . . . 32 2.13 Sensing matrix of the vertex degrees of freedom . . . . . . . . . . . . . . 33 2.14 OLTFs of the vertex length degrees of freedom . . . . . . . . . . . . . . . 35 2.15 Feedforward subtraction diagram . . . . . . . . . . . . . . . . . . . . . . . 36 2.16 PRC length coupling into differential arm length . . . . . . . . . . . . . . 37 2.17 Beamsplitter motion coupling into differential arm length . . . . . . . . 38 2.18 SRC length coupling into differential arm length . . . . . . . . . . . . . . 39 2.19 Diagram of hard and soft cavity optic motions. . . . . . . . . . . . . . . . 42 2.20 Open-loop transfer functions for angular control of the differential hard arm mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.21 Open-loop transfer functions for angular control of the beamsplitter. . 46 ix 2.22 Open-loop transfer functions for angular control of the signal recycling cavity pointing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.23 Diagram of the Advanced LIGO quadruple suspension. . . . . . . . . . . 50 3.1 GW photodiode auto- and cross-spectral densities from O1. . . . . . . . 56 3.2 Diagram of the signal recycling mirror holder. . . . . . . . . . . . . . . . 64 3.3 Budget of frequency noise in H1. . . . . . . . . . . . . . . . . . . . . . . . 69 3.4 Frequency noise coupling into the differential arm length readout. . . 71 3.5 Relative intensity noise at the interferometer input. . . . . . . . . . . . 72 3.6 Intensity noise coupling into the differential arm length readout, in terms of input RIN into antisymmetric-port RIN. . . . . . . . . . . . . . . 73 3.7 Noises associated with the test mass electrostatic actuator. . . . . . . . 74 3.8 Freerunning and residual length noises for the power recycling cavity. 79 3.9 Freerunning and residual length noises for the signal recycling cavity. 80 3.10 Freerunning and residual length noises for the Michelson degree of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.11 Noise budget of Advanced LIGO fundamental noises. . . . . . . . . . . . 82 3.12 Noise budget of Advanced LIGO Hanford at the start of the first observing run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Spectrogram of the GW150914 signal in the H1 detector. . . . . . . . . 85 4.2 Advanced LIGO H1 strain spectra during the three events GW150914, LVT151012, and GW151226. . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.1 Simplified diagram of the Advanced LIGO differential arm length sensing and control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 Advanced LIGO differential arm length optomechanical plants . . . . . 90 5.3 Open- and closed-loop transfer functions of the differential arm length loop used in the simulations in this chapter. . . . . . . . . . . . . . . . . 97 5.5 Normalized Cramér–Rao bounds on the covariance matrix for a singlefrequency transfer function estimate of a one-pole system, assuming a fixed excitation amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.6 Normalized Cramér–Rao bounds on the covariance matrix for a singlefrequency transfer function estimate of a one-pole system, assuming a fixed excitation SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.7 Posterior for optimal calibration line placement for NS/NS-detuned operation at 125 W. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.8 Frequency-dependent statistical errors in the NS/NS-optimized detuned interferometer response function for an optimized calibration line scheme.107 x 5.9 Frequency-dependent statistical errors in an O2-like response function for a by-hand choice of calibration lines. . . . . . . . . . . . . . . . . 109 5.10 Frequency-dependent statistical errors in an O2-like response function for an optimized choice of calibration lines. . . . . . . . . . . . . . . 110 6.1 Example acceleration time series for dark matter interactions with Advanced LIGO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6.2 Example acceleration spectra for dark matter interactions with Advanced LIGO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3 Event rates for Newtonian dark matter interaction with laser interferometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.4 Event rates for Yukawa dark matter interaction with Advanced LIGO. 117 6.5 Event rates for Yukawa dark matter interaction with LISA. . . . . . . . 117 7.1 Expected frequency noise coupling with and without a signal recycling mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 7.2 Expected intensity noise coupling with and without a signal recycling mirror. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.3 Expected frequency noise coupling for a possible 40 km detector. . . . . 123 A.1 Anatomy of a generalized servo loop. . . . . . . . . . . . . . . . . . . . . . 126 B.1 Mean versus median averaging for the H1 OMC null stream. . . . . . . 135 B.2 Histogram of ASD values for the H1 OMC null stream at 100 Hz. . . . . 135 E.1 Posterior pdf of Advanced LIGO test mass loss angles, as determined from end Y ringdown data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 xi LIST OF TABLES Number Page 2.1 Optomechanical parameters for the Advanced LIGO Hanford detector during O1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Paramters for aLIGO H1 laser frequency stabilization. . . . . . . . . . . 19 2.3 Paramters for differential arm length sensing and control during O1 . 30 2.4 Parameters for vertex length sensing and control during O1 . . . . . . 33 2.5 Sensors and bandwidths for angular control during O1 . . . . . . . . . . 41 5.1 Parameters for the Advanced LIGO optomechanical plant. . . . . . . . . 91 B.1 Median bias factor b (to four decimal places) for N non-overlapping segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 E.1 Data for determining coating and substrate losses via end Y test mass ringdowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 xii Published Content and Contributions 1. E. D. Hall, V. V. Frolov, T. Callister, H. Müller, M. Pospelov, and R. X. Adhikari. Laser interferometers as dark matter detectors. ArXiv e-prints (2016). E.D.H. performed numerical simulations for Newtonian and Yukawa event rates and wrote portions of the manuscript. [arXiv:1605.01103]. 2. Evan Hall for the LIGO Scientific Collaboration. Laser Frequency and Intensity Stabilization for Advanced LIGO. 18th Coherent Laser Radar Conference (CLRC 2016). Cooperative Institute for Research in Environmental Sciences (CIRES), 2016, pp. 197–200. isbn: 9781510828544. E.D.H. wrote this overview on behalf of the LIGO Scientific Collaboration. [URL: https://clrccires.colorado.edu/4pagesummaries/W/W2.pdf]. 3. D. V. Martynov, E. D. Hall, B. P. Abbott, R. Abbott, T. D. Abbott, C. Adams, R. X. Adhikari, R. A. Anderson, S. B. Anderson, K. Arai, et al. Sensitivity of the Advanced LIGO detectors at the beginning of gravitational wave astronomy. Phys. Rev. D 93 (11) (2016), p. 112004. E.D.H. contributed a high-frequency strain noise budget and a brief discussion of laser frequency noise. [DOI: 10.1103/PhysRevD.93.112004]. 1 1 General relativity and gravitational radiation In this chapter, we’ll review some basics of general relativity and gravitational radiation. Largely we will follow Creighton and Anderson.1 A good overview of general relativity is given by Zee.2 1.1 Basic equations Geodesic equation A central postulate in general relativity is that massive particles move to extremize their proper time τ. In other words, the spacetime trajectory X µ (τ) is found by extremizing the action )1/2 ∫s B ∫τB ( dX α dX β S = ds = dτ g αβ . dτ dτ (1.1) τA sA The result of this extremization is the geodesic equation ρ ν d2 X µ µ dX dX = Γ , νρ dτ dτ dτ2 where 1 µ Γνρ = gµλ 2 ( ∂ g ρλ ∂ g σλ ∂ g ρσ + − σ ∂xρ ∂x ∂xλ (1.2) ) (1.3) is the connection coefficient. Additional forces (from electromagnetism, elastodynamics, and so on) go on the right-hand side of equation 1.2. 2 Equation of geodesic deviation How can one tell whether a particular metric describes flat spacetime or a spacetime with gravity? One prescription is to start with two test masses, initially separated by a small spacetime vector ζµ , and set them in motion with some four-velocity U ρ = dX ρ /dτ. Then the relative acceleration of the two masses is given by the equa- tion of geodesic deviation: D2 ζµ = −R µ νρσU νU ρ ζσ , Dτ 2 ( ) where D2 ζµ /Dτ2 = U λ ∇λ U σ ∇σ ζµ and µ µ R µ νρσ = 2 ∂[ρ Γσ]ν − 2Γλν[ρ Γσ]λ (1.4) (1.5) is the Riemann tensor. If D2 ζµ /Dτ2 is nonzero, the spacetime is curved. Geodesic deviation is the principle that underpins gravitational wave detection. Einstein field equation The relationship between spacetime curvature and matter/energy is given by the Einstein field equation, which tells us how to calculate the metric tensor g αβ given a certain stress-energy tensor Tαβ : G αβ = 8π Tαβ , c4 (1.6) where R αβ = R γ αγβ R = Rγγ G αβ = R αβ − 12 g αβ R (Ricci tensor) (1.7) (scalar curvature) (1.8) (Einstein tensor). (1.9) In the following sections we will see that in empty space (Tαβ = 0), the Einstein equation can be linearized and then admits plane wave solutions, which are gravitational waves. Observations of type Ia supernovae and of anisotropies of the cosmic microwave background indicate that (at least on cosmological scales) the behavior of “empty” space is consistent with the presence of a constant, nonzero stress-energy Tαβ = −Λ g αβ (so-called “dark energy”), which is about 70 % of the closure density of the universe.3 3 1.2 Weak-field limit Many experimental tests of general relativity (including the detection of gravitational waves from distant astrophysical sources) operate in a regime where the spacetime curvature is small. In this so-called weak-field limit, the metric g αβ is approximately the Minkowski metric ηαβ with a perturbation h αβ satisfying | h αβ | ≪ |ηαβ |. In this regime, terms involving products of h αβ with itself are ignored. In the weak-field limit, the spacelike components of the geodesic deviation equation are ζ̈ i = −R i 0 j0 ζ j . (1.10) This shows that the components R i 0 j0 of the Riemann tensor manifest physically as tidal accelerations. For example, in the case of a spacetime sourced by a Newtonian potential Φ(x i ), these components R i 0 j0 are the tidal field − ∂2 Φ/ ∂x i ∂x j . Turning this statement around, we can say that one can measure the Riemann tensor (and hence the curvature of spacetime) by measuring a tidal acceleration. 1.3 Gravitational radiation In the weak-field limit, the Einstein field equations can be recast as a wave equation in h αβ , sourced by some stress–energy tensor Tαβ . Solving this wave equation in vacuum (Tαβ = 0) yields a class of plane wave solutions, called gravitational waves. For a wave travelling along the z axis, the weak-field metric perturbation has the form   0 0 0 0   0 h h × 0 +   h αβ =  , 0 h × − h + 0   0 0 0 0 (1.11) where the coefficients h + and h × are (potentially) time-dependent.a Equivalently, the line element for this metric is ds2 = − c2 dt2 + (1 + h + )dx2 + (1 − h + )dy2 + 2h × dx dy + dz2 . (1.12) a Note that there are only two independent components in this metric, despite the fact that the metric can have up to ten independent components in general. In this case, eight of these ten components are eliminated when fixing the gauge. This gauge is referred to as the transverse traceless (TT) gauge, and freely falling masses have fixed coordinates in this gauge, even in the presence of a time-varying metric. 4 In this case, the geodesic deviation equations are ( ) ζ̈ x = 12 ḧ + ζ x + ḧ × ζ y ( ) ζ̈ y = 12 ḧ × ζ x − ḧ + ζ y ζ̈ z = 0, (1.13a) (1.13b) (1.13c) where the baseline separation vector is ζ = ζ x x̂ + ζ y ŷ + ζ z ẑ. These equations show explicitly that a gravitational wave exerts a tidal force on any matter that it passes through. 1.4 Terrestrial detection Terrestrial detection efforts revolve around trying to make very sensitive tidal force measurements in the audio band. Bars Resonant bar detectors are the oldest class of gravitational wave detection experiments.1 In these experiments, a high-Q object (often a bar, and often made of high-quality aluminum) is used to detect the tidal force of a passing gravitational wave. A wave with appropriate frequency content can pump energy into one of the bar’s normal modes, setting the ends of the bar in motion with a strain that is much greater (by a factor Q ) compared to the intrinsic strain of the gravitational wave. Modern resonant bar detectors are able to achieve narrow-band strain sensitivities of order 10−21 /Hz1/2 .1,4 Planets and moons Just like a metal bar, a celestial body such as a planet or moon will respond to the stress of a passing gravitational wave. The response produces local seismic disturbances at the body’s surface,7 which can be measured using a network of seismometers. Coughlin and Harms used data from seismometer networks on the Earth8 and the Moon9 to place upper limits on the stochastic GW background ΩGW from 0.1 to 1 Hz, yielding a limit ΩGW < 1.2 × 105 . 5 10−17 Bars eLIGO aLIGO O1 10−18 aLIGO design DECIGO LISA £ ¤ ASD of strain 1/Hz1/2 10−19 10−20 10−21 10−22 10−23 10−24 10−5 10−4 10−3 10−2 10−1 100 101 102 103 104 Frequency [Hz] Figure 1.1: Strain sensitivities for some past, current, and future gravitational wave experiments. “Bars” refers to the joint AURIGA–EXPLORER–NAUTILUS– Virgo run described by Acernese et al.4 “DECIGO” refers to a proposed space-based laser interferometer experiment with a 1000 km baseline.5 “LISA” refers to a proposed space-based laser interferometer experiment with a 5 × 106 km baseline.6 Laser interferometers An optical interferometer monitors the relative displacement (and hence relative acceleration) of multiple freefalling (or quasi-freefalling) test masses. In the case of a simple Michelson interferometer, one mirror (test mass) is displaced by an amount ζ x x̂ from a beamsplitter, and another is displaced an amount ζ y ŷ from a beamsplitter. One port of the beamsplitter is illuminated with laser light of wavelength λ0 , and a photodetector is used to monitor the amount of light exiting the other port. If the relative displacement is zero (ζ x = ζ y = ζ), the interferometer’s antisymmetric port will be dark, since the round-trip phases ϕ x and ϕ y are equal. 6 Explicitly, ∫ζ/c ϕ x = 2c 0 2π 4π dt = λ0 λ ∫ζ dx = 0 4π ζ, λ (1.14) and the calculation for ϕ y yields the same answer. We now consider what happens when a gravitational wave passes through the detector. We are interested in waves with frequency f of less than a few kilohertz, so the period T = 1/ f is on the order of milliseconds or more. On the other hand, the light travel time in a kilometer-scale Michelson is tens of microseconds. Therefore, the response of the interferometer to the wave can be treated adiabatically. Suppose at some particular time the GW strain is h + . We again want to compute the phases ϕ x and ϕ y of the light that is sent into the interferometer in the presence of this strain. For light traveling along the x arm, dt and dx are related by 0 = − c2 dt2 + (1 + h + )dx2 because light always travels along a null path. Therefore, the phase accumulated in the x arm isb ∫ζ/c ϕx = 2 0 2π c 4π dt = λ0 λ0 ∫ζ √ 0 and similarly, 4π 1 + h + dx ≃ λ0 ∫ζ ( 0 ) ( ) h+ 4π h+ 1+ dx = 1+ ζ, 2 λ0 2 ( ) 4π h+ ϕy = 1− ζ, λ0 2 (1.15) (1.16) so that 4π h + ζ. (1.17) λ0 This shows that a Michelson interferometer will register a phase shift ∆ϕ that is ∆ϕ = ϕ x − ϕ y = directly proportional to both the applied spacetime strain h + and the length of the interferometer’s baseline ζ. We assumed the light was sent into the interferometer after the strain had already been applied, so that the wavelength is the same in both arms. However, any light already circulating in the interferometer as the strain is applied will be gravitationally frequency-shifted, with the light in one arm being redshifted and the light in the other arm being blueshifted.10 As the light in the arms returns to the beamsplitter, a phase difference accumulates, causing power to appear at the dark port. This is the resolution to the common question of how one can use light to detect gravitational waves, given that gravitational waves cause redshift of the light in the arms. b Even in the presence of nonzero strain, the limits of the integration are still 0 and ζ, because the masses are assumed to be freely falling, and therefore their coordinates do not change in the TT gauge. 7 1.5 Other detection methods Pulsar timing Millisecond pulsars are excellent clocks that can be used for GW detection.11 If a GW passes between the line of sight from a pulsar to a radio telescope, then the arrival times of the pulses will be perturbed. There are many non-GW effects which could perturb the pulse arrival time for a particular pulsar, so in practice many pulsars are observed simultaneously. This makes it possible to extract the correlated arrival time perturbation due to the GW. Pulsar timing is particularly well-suited to measurement of ΩGW ( f ) on year-long timescales; the Parkes Pulsar Timing Array has been able so far to set a limit ΩGW (1 /yr) < 2.3 × 10−10 assuming ΩGW ( f ) ∝ f 0.5 .12 Cosmic microwave background polarization A detection of gravitational waves from the early universe can be made by careful measurement of the cosmic microwave background (CMB) polarization.13 If the inflationary paradigm is correct, then the present observable universe evolved from a small, causally connected region that was blown up by a large factor (∼10 to ∼60 e-foldings) less than 10−32 s after the big bang. Consequently, quantum fluctua- tions in the fields present during the inflationary period—for example, the inflaton field and the gravitational field—were similarly blown up and should appear today as perturbations to the temperature and polarization of the CMB on large angular scales. While several different fields can produce large-angle curl-free (E -mode) perturbations in the CMB polarization in the early universe, only the gravitational field can produce large-angle divergence-free (B-mode) CMB perturbations. Therefore, detecting primordial B-modes in the CMB would provide evidence for the existence of gravitational waves in the early universe. Detecting primordial B-modes is the current aim of several ground-based, balloon-borne, and space-based CMB polarimetry experiments. 8 2 The Advanced LIGO interferometers: design and control This chapter discusses the optical topology of the Advanced LIGO interferometers, and how the interferometer lengths, angles, and laser are controlled. This thesis will not discuss the lock acquisition system (green auxiliary lasers for the arms and third-harmonic heterodyne sensing for the vertex)14 used to bring the interferometer onto resonance from its initial uncontrolled state. This system has already been discussed in detail by Staley15 and Martynov.16 We will make heavy use of the analytical formulas found by Sigg17 and Izumi and Sigg18–21 for length readout and control of resonant interferometers. 2.1 Topology overview This section will review Michelson and Fabry–Pérot interferometer topologies, and how these are combined to form a dual-recycled Fabry–Pérot Michelson interferometer. Michelson interferometer Consider a Michelson interferometer with two reflective test masses and a 50 % beamsplitter (figure 2.1). Suppose the input (symmetric) port of the Michelson is illuminated with a laser with frequency ω0 /2π and field amplitude E 0 . If ϕX = 2ω0 ℓX /c and ϕY = 2ω0 ℓY /c are the round-trip phases from the beamsplitter to the test masses and back, then the 9 Figure 2.1: Diagram of a Michelson interferometer. A laser beam is injected into the symmetric (bright) port, and the differential arm length signal is read out at the antisymmetric (dark) port. field at the antisymmetric port is ) E 0 ( iϕX iϕ Y E AS = e −e 2 = i E 0 eiϕ+ sin ϕ− , (2.1) (2.2) ¯ ¯2 with ϕ± = (ϕX ± ϕY )/2. Correspondingly, the power PAS = ¯E AS ¯ at the antisymmet- ric port is PAS = P0 sin2 ϕ− . (2.3) Implicit in this equation is the statement that common-mode phase fluctuations do not make a signal at the antisymmetric port. This common-mode rejection is crucial for interferometric gravitational wave detection, since all realistic lasers have phase noise that is many orders of magnitude higher than the phase fluctuation induced by a passing gravitational wave. How is the anti-symmetric fringe condition maintained? One possibility is to servo the laser frequency so that PAS = P0 /2 (the half-fringe condition). This method is simple, but has the disadvantage that fluctuations in P0 appear directly at the antisymmetric port. 10 Another possibility (closer to what is actually employed) is to modulate the laser at some radio frequency Ω/2π and then operate the Michelson at a dark fringe (PAS = 0). This requires choosing ℓX ̸= ℓY by some macroscopic amount (that is, by many multiples of the laser wavelength), so that the transmission of light from the input port to the anti-symmetric port becomes frequency-dependent. In the field of gravitational wave detection, this intentional length offset is called the Schnupp asymmetry.a Writing ϕ± (ω) = 2ω ℓ± /c, we see that E AS (ω) = i E 0 (ω) e2iω ℓ+ /c sin (2ω ℓ− /c) . (2.4) We write ℓ− = ℓ− + δ ℓ− , where ℓ− is the Schnupp asymmetry (i.e., the macroscopic part of the differential length), and δ ℓ− is the microscopic detuning. We assume the Michelson is operated at the dark fringe, so we are trying to keep δ ℓ− at 0. The field incident on the beamsplitter is E0 e iω0 t iΓ cos Ω t e ≃ E0 e iω 0 t ( ) iΓ iΩ t iΓ −iΩ t 1+ e + e , 2 2 (2.5) where Γ is the modulation index and Ω is the angular modulation frequency. After some algebra (see section C.3), it can be shown that, to linear order in δϕ− , the demodulated rf power at the antisymmetric port at Ω is PAS = P0 Γ sin ϕ− δϕ− , (2.6) where ϕ− = 2ω ℓ− /c is the macroscopic portion of the differential phase. What is the phase sensitivity of a Michelson interferometer? In the case of either dc or rf readout, it is set by the power incident on the beamsplitter. In the case of dc readout, if the antisymmetric port is kept nearly dark, then PAS ≃ ¯ P0 ϕ2 , so that the optical gain is s dc = ∂PAS / ∂ϕ− ¯ = 2P0 ϕ0 , where ϕ0 is the static − ϕ0 differential phase offset for the carrier in the arms. The shot noise of the light in the antisymmetric port is S 1/2 ( f ) = (2hν0 PAS )1/2 = (2hν0 P0 )1/2 ϕ0 , and hence the PP equivalent phase noise is (hν0 /2P0 )1/2 .23 In the case of rf readout, the optical gain is s rf = P0 Γ sin ϕ− , and the shot noise (which comes from the two sidebands) is [(3/2)hν0 P0 ]1/2 Γ sin ϕ− ; the shot-noise-limited a This is the same basic principle that underlies Fourier transform spectroscopy, which has been used (among other things) to verify the thermal nature of the cosmic microwave background.22 11 Figure 2.2: Diagram of a Fabry–Pérot interferometer. The photodiode may also be placed in so as to monitor the reflected light, rather than the transmitted light. phase sensitivity is then [3hν0 /2P0 ]1/2 , independent of both Γ and ϕ− .b We can refer this noise to strain via the relation δϕ = (π/λ0 )L δ h. This results in a shot-noiselimited strain ASD of 1 S (shot) ( f )1/2 = hh πL ( ) 3hcλ0 1/2 . 2P0 (2.7) For P0 = 1 W, λ0 = 1064 nm, and L = 4 km, this results in a strain of 4 × 10−20 /Hz1/2 . This strain sensitivity is good, but not good enough for regular detection of gravitational wave events. Improving this sensitivity requires either increasing the laser power by many orders of magnitude or augmenting the optical topology to improve the shot-noise-limited SNR. In the next section we show that the sensitivity can be greatly enhanced through the use of Fabry–Pérot interferometry. Fabry–Pérot interferometer The basic definitions for a Fabry–Pérot interferometer are given in appendix C. This appendix also reviews the standard control scheme [the Pound–Drever–Hall (PDH) technique] for locking a laser to a cavity, or vice versa. We consider a two-mirror Fabry–Pérot interferometer with length L (figure 2.2). For simplicity we assume the input mirror has transmissivity T , and the output mirror has zero transmissivity. Therefore, the cavity is overcoupled, with a pole f p = cT/8πL. We now compute the shot-noise-limited sensitivity of this cavity when it is kept resonant using PDH reflection locking. Given a modulation depth Γ, the optical gain (in watts per meter) of the PDH readout isc / 4P0 J0 (Γ)J1 (Γ) f p c / s( f ) = × . L λ0 1 + i f fp (2.8) The shot noise on the photodiode is [ ] S (shot) ( f ) = 4hν0 P0 (1 − v) J0 (Γ)2 + 3J1 (Γ)2 . PP (2.9) b The factor of 3 (rather than 2) in these rf shot noise formulas arises from the cyclostationary nature of the noise in the demodulated sidebands.24,25 c See appendix C. 12 Assuming the visibility is perfect (v = 1), the shot-noise limited strain ASD is therefore 1 S (shot) ( f )1/2 = × hh L S (shot) ( f )1/2 PP | s( f )| ( ) ¯ 1 3hλ0 1/2 ¯¯ = fp + i f ¯ . 2 cP0 (2.10) For L = 4 km and f p = 42 Hz, we have S (shot) ( f ≪ f p )1/2 = 2 × 10−24 /Hz1/2 , an improvehh p /p ment of c 2πL f p = 4F/ 2π ≃ 400 over the Michelson sensitivity as given by (2.7). Above the cavity pole, the sensitivity rises like f , crossing the Michelson sensitivity /p at f = c 2πL ≃ 17 kHz.d It is therefore highly advantageous to use optical cavities when using a laser to sense small audio-band strains. In the next section, we examine the Fabry–Pérotenhanced Michelson topology that is used in Advanced LIGO and other gravitationalwave interferometers. Fabry–Pérot Michelson interferometer with recycling The Advanced LIGO instruments are dual-recycled Fabry–Pérot Michelson interferometers, as shown in figure 2.3. This means that they are Michelson interferometers whose arms are Fabry–Pérot cavities. Additionally, a power-recycling mirror is placed between the laser and the beamsplitter in order to enhance the amount of power circulating in the Fabry–Pérot arms. Finally, a signal-recycling mirror is placed between the beamsplitter and the antisymmetric port readout to tune the bandwidth of the instrument. A dual-recycled Fabry–Pérot Michelson has five length degrees of freedom: LX + LY 2 LX − LY L− = 2 L+ = xIX − xB yIY − yB + 2 2 xIX − xB yIY − yB LM = − 2 2 xIX − xB yIY − yB + . L S = − yS + yB + 2 2 L P = − xP + xB + (2.11) (2.12) (2.13) (2.14) (2.15) d We note in passing that the shot-noise-limited sensitivity of a Fabry–Pérot cavity is independent of f p for f > f p ; in other words, there is no penalty in terms of shot-noise-limited sensitivity for choosing f p to be as small as possible.26 13 These are, respectively, the common-mode arm length (CARM), the differential arm length (DARM), the power recycling cavity (PRC) length, the Michelson length (MICH), and the signal-recycling cavity (SRC) length.e;f We now want to answer the question: how do we choose the interferometric lengths and the optic reflectivities? We should choose the end test masses to be highly reflective, since there is no benefit in letting light leak out the end mirrors, other than the small amount (about 1 W) required for diagnostic and control purposes. In Advanced LIGO, we have Te = 4 ppm. The input test mass transmissivity Ti and signal recycling mirror transmissivity Ts are chosen jointly to set the interferometer’s signal bandwidth and to optimize a number of technical effects; we discuss these transmissivity choices later. We now discuss how to choose the power-recycling mirror transmissivity Tp . We want to choose the mirror transmissivity so as to provide the lowest possible strainreferred shot noise, which means maximizing the amount of power entering the arms. For the time being, suppose we have chosen some fiducial input test mass transmissivity Ti , and suppose each arm has a round-trip loss ηa of order 100 ppm. With r i ≃ 1 − Ti /2 − ηi /2 and r e ≃ 1 − ηe /2, the arm reflectivity is18 ra = r e (1 − ηi ) − r i Ti − ηa 2ηa ≃ ≃ 1− , 1 − ri re T i + ηa Ti e Note that the differential arm length is sometimes defined as L (2.16) − = L X − L Y , particularly for the purposes of noise budgeting and data analysis. In particular, the equivalent GW strain estimate stored in the science data is h = (L X − L Y )/L, where L is the nominal average arm length. f Why are there only five interferometric degrees of freedom if a dual-recycled Fabry–Pérot Michelson interferometer has seven mirrors? If each mirror is constrained to lie in the x y plane, then there are fourteen translational degrees of freedom (two for each mirror). Moving any of the seven optics transverse to their surface normals will obviously produce no signal, so this eliminates seven degrees of freedom. The other two non-interferometric degrees of freedom involve the simultaneous motion of multiple optics: 1. The four test masses move away from the beamsplitter along the beamline directions (which has no effect on L + , L − , or L M ), and the power- and signal-recycling mirrors move toward the beamsplitter in order to null the change in L P and L S . 2. The IX and EX masses move away from the beamsplitter, and the IY and EY masses move toward the beamsplitter. This has no effect on L + L − , L p , or L s , but it lengthens L M . Therefore, the beamsplitter and the recycling mirrors are moved in order to cancel the effect in LM. 14 where ηa = ηi + ηe . Then ¯ ¯ ¯ t p ¯2 Pb 2 ¯ = g p = ¯¯ P0 1 − rp ra ¯ Tp ≃ , (Tp /2 + ηp /2 + 2ηa /Ti )2 (2.17a) (2.17b) where ηp is the round-trip PRC loss.g If ηp ≪ 4ηa /Ti , the PRC loss is negligible and hence the PRC gain Pb /P0 is maximized by choosing Tp = 4ηa /Ti , (2.18) giving g2p = Ti /4ηa . We now discuss how to choose Ti and Ts . In a non-signal-recycled interferometer, the quantum noise limit depends crucially on the choice of arm finesse. However, a signal-recycled interferometer has a quantum noise limit that is determined jointly by the arm finesse and the signal recycling mirror transmissivity. Therefore, the arm finesse may be determined from any number of other criteria, and the signal recycling mirror transmissivity may be adjusted to give the desired quantum noise curve. Some of these finesse criteria are as follows. • The coupling of Michelson motion into the antisymmetric port is given by 1/G a = π/2F.27 To ensure that thermal noise from the beamsplitter suspen- sion contributes negligibly to the differential arm length readout, we should choose F ≥ 350 for Advanced LIGO.28 • The coupling of low-frequency SRC length motion into the antisymmetric port is proportional to the stored arm power, the arm finesse, and the dc readout offset.27 To ensure that thermal noise from the signal recycling mirror suspension contributes negligibly to the differential arm length readout, we should choose F ≤ 625 for Advanced LIGO.28 • A lower finesse arm easier to lock, since it spends more time crossing each fringe. • Lower power stored in the arms leads to fewer parametric instabilities. The Advanced LIGO arms are chosen to have a finesse of 440, corresponding to an input test mass transmissivity of Ti = 1.4 %. This means that each arm has a natural bandwidth (cavity pole) f a = cTi /8πL = 42 Hz. For the final Advanced LIGO design, the signal recycling mirror is chosen to have a transmissivity Ts = 20 %, corresponding to a bandwidth (differential arm pole) f − ≃ f a × (1 + r s )/(1 − r s ) = 750 Hz, g This follows by writing r p= √ 1 − Tp − ηp ≃ 1 − Tp /2 − ηp /2. 15 where r s = √ 1 − Ts . However, for lower power operation (as during the first observ- ing run), the transmissivity is instead Ts = 37 %, giving a bandwidth f − ≃ 365 Hz. The exact expressions for the common-mode (+) and differential (−) arm poles are18 [ ] 1 + ri rp c 2π f + = (2.19) ln 2L + r e r i + r e r p (1 − ηi ) [ ] 1 − ri rs c . 2π f − = ln (2.20) 2L + r e r i − r e r s (1 − ηi ) Fixing Ti also fixes the value of the PRM transmissivity required to achieve maximal power buildup (that is, critical carrier coupling) in the interferometer, according to equation 2.17b. However, choosing critical carrier coupling means that certain reflection error signals that depend on the beat of the carrier against the sidebands will vanish (the 45 MHz reflection angular signals are one such example— see section 2.5). Therefore, the PRM transmissivity is chosen to be Tp = 3.0 %, which slightly overcouples the carrier. Finally, we consider how to choose the macroscopic cavity lengths. The macroscopic lengths of the common-mode and differential arm lengths should be obvious: the common-mode length should be as long as possible, so as to get the best strain sensitivity, and the differential arm length should be zero, so that the interferometer is balanced. For the vertex lengths, the rough goal is to have one set of sidebands (at frequency f 1 ) resonant in the PRC and another (at frequency f 2 ) resonant in the SRC, with both sidebands nonresonant in the arms. To achieve nonresonance in the arms, the sideband frequency must be significantly different from a multiple of c/2L + = 37.5 kHz. Once the sideband frequencies are chosen, the macroscopic PRC and SRC lengths (and the Schnupp asymmetry for the Michelson degree of freedom) must be chosen to give the appropriate resonance condition. In particular, the PRC length ℓp must be a half-integer multiple of c/2 f 1 , so that the carrier and the sideband are simultaneously resonant. If the rf modulation is applied before the input modecleaner (as is the case in Advanced LIGO), the modecleaner length must also be a multiple of c/2 f 1 . The SRC length ℓs is then chosen to be a half-integer multiple of c/2 f 2 , such that the f 1 sidebands do not also resonate. Finally, the Schnupp asymmetry ℓm is chosen to critically couple the f 2 sidebands into the SRC.27 Numerical values for the Advanced LIGO optical configuration are given in table 2.1, and an overview diagram is given in figure 2.3. 16 9, 45 end test mass (4 ppm) (from input modecleaner) 9, 45 input test mass (1.4 %) compensation plate (100 %) beamsplitter (50 %) power recycling mirror (3.0 %) 9, 45 36, 45 signal recycling mirror (⁕) 45 36, 45 Figure 2.3: Overview of the Advanced LIGO interferometer. Optics and photodetectors inside the dashed box are in vacuum and seismically isolated. Numbers next to photodetectors indicate resonant rf detection at those frequencies. Additionally, each resonant rf length sensor has a corresponding broadband rf detector placed in air (for detecting frequencies from 9 to 135 MHz). Numbers next to optics indicate transmissivities. The signal recycling mirror has a transmissivity of 37 % for O1 and O2; the design value, for use at full laser power (125 W), is 20 %. fused-silica reference cavity 9.1 MHz (to test masses) 24 MHz 2 W, 1064 nm seed NPRO suspended input modecleaner ×5 45.5 MHz 35 W single-pass amplifier power recycling mirror (from AOM) output modecleaner signal recycling mirror input test masses (to sum point) main interferometer gravitational wave readout end test masses (from AOM) (to sum point) tion (green lines). The differential arm length control and the strain readout are shown with black lines. Some liberties have been taken with folding and pickoff mirrors. Components for sensing and control of the injection-locked amplifier, pre-modecleaner, beamsplitter, recycling mirrors, and output modecleaner are not shown. The outermost intensity stabilization path (feeding the transmitted arm powers to the inner servo error point) was not used during the first observing run. Symbols: photodiode; rf oscillator; electro-optic acousto-optic modulator; mixer. modulator; Figure 2.4: Simplified diagram of the Advanced LIGO laser system, including frequency stabilization (blue lines) and intensity stabiliza- steel-body pre-modecleaner (from arms) 21 MHz 220 W injection-locked amplifier 17 18 Arm length 3994.5 m PRC length 55 m Michelson length 3m Schnupp asymmetry 8 cm SRC length 56 m Input modecleaner length29 33 m 30 Output modecleaner length 1.132 m End test mass transmission 4 ppm Input test mass transmission 1.4 % Power recycling mirror transmission 3.0 % Signal recycling mirror transmission 37 % Arm loss 31 100(20) ppm Interferometer rf modulation frequency 1 9.100 230 MHz Interferometer rf modulation frequency 2 45.501 150 MHz IMC rf modulation frequency 24.078 360 MHz OMC dither frequency 4.1 kHz Interferometer rf modulation depth 1 0.22 rad Interferometer rf modulation depth 2 0.28 rad IMC rf modulation depth 0.01 rad Laser wavelength 1064 nm Interferometer input power 19 to 21 W Carrier power recycling gain 38 to 42 W/W Carrier arm gain Carrier visibility 280 W/W > 97 % Table 2.1: Optomechanical parameters for the Advanced LIGO Hanford detector during the first observing run. The parameters for the Livingston detector are similar. 2.2 Laser frequency control Overview Reducing the laser frequency noise to an astrophysically competitive level is challenging. As said above, the interferometer’s Michelson topology should reject commonmode noises such as laser frequency noise. However, this common-mode rejection is 19 Reference cavity length32 20.3 cm 32 290 μm Reference cavity pole32 30 kHz Reference cavity servo bandwidth 200 kHz Input modecleaner round-trip length33 32.9 m Reference cavity spot size Input modecleaner spot size34 2–3 mm Input modecleaner pole 9 kHz Input modecleaner servo bandwidth 40 kHz Common-mode arm length33 3994.5 m 34 5–6 cm Common-mode cavity pole 0.63 Hz Common-mode spot size Common-mode servo bandwidth 15 kHz Table 2.2: Paramters for aLIGO H1 laser frequency stabilization. finite; in practice, it is a factor of a few hundred. To make the laser noise contribute negligibly to the interferometer’s GW strain readout, a phase noise of 10−8 rad/Hz1/2 at 100 Hz is required at the interferometer’s input.35 (The common-mode cavity then provides a factor of ∼ 100 suppression for light circulating in the arms.) On the other hand, the fractional phase fluctuation of a good solid-state laser such as a 1064 nm Nd:YAG NPRO has a freerunning noise of about 100 Hz/Hz1/2 at 100 Hz,36 which amounts to a phase fluctuation of 1 rad/Hz1/2 . How can we achieve eight orders of magnitude of phase noise suppression below 1 kHz? First, we need an optical reference cavity whose phase noise is at least as good as the phase noise requirement. Such a length reference is handily furnished by the interferometer’s common-mode arm length: within the gravitational wave band (10 Hz to 7 kHz), the displacement noise of this degree of freedom should be limited by radiation pressure noise, coating thermal noise, squeezed film damping, and perhaps a few other displacement noises which nonetheless are too small to cause the resulting laser phase noise to contaminate the differential arm length readout. How do we build a loop to frequency-stabilize the laser to the common-mode arm length? We might imagine building a simple frequency-locking loop in which the freerunning laser is directly locked to the common-mode arm length. However, this loop would need its final unity-gain frequency to be 1 MHz or more. Such a Magnitude [Hz/Hz] 20 102 101 100 10−1 104 105 106 105 106 90◦ Phase 45◦ 0◦ −45◦ −90◦ 104 Frequency [Hz] Figure 2.5: Open-loop transfer function of the reference cavity stabilization servo, with no feedback from the input modecleaner or the interferometer. The dip in the transfer function around 26 kHz arises from the actuation crossover between the laser PZT and the broadband EOM. high UGF is difficult to achieve, since a 4 km cavity has a phase delay of L/c = 13 μs in its PDH signal.37 Instead, Advanced LIGO (as with other initial/advanced gravitational-wave laser interferometers) employs a multi-cavity stabilization approach: the laser is first stabilized to a table-top reference cavity using a loop with a bandwidth of hundreds of kilohertz, then to a suspended cavity (the input modecleaner) with a bandwidth around 50 kHz, and finally to the common-mode arm length with a bandwidth of about 20 kHz. Table 2.2 gives parameters relevant to the laser frequency stabilization, and a diagram of the frequency (and intensity) stabilization scheme is shown in figure 2.4. Reference cavity stabilization The goal of the reference cavity loop is to suppress the freerunning frequency noise ( ) ( / ) of the NPRO—about 100 Hz/Hz1/2 × 100 Hz f —down to the length noise of a 21 20.3 cm fused-silica reference cavity. The noise performance of this stabilization scheme was investigated at length by Chalermsongsak et al.32 This cavity is 20.3 cm long, has a finesse of roughly 104 , and is designed to be critically coupled. It is suspended inside a small vacuum chamber that sits on the main pre-stabilized laser table. In the absence of scatter, rf amplitude modulation, excess intensity noise, and so on, this design should produce a cavity frequency noise that is limited by the Brownian noise of the mirror coatings—about ( ) ( / )1/2 6 mHz/Hz1/2 × 100 Hz f . About 20 mW of light from the main laser beam is incident on the cavity. 21 MHz phase sidebands are applied with a resonant electro-optic modulator, and the resulting beat note is sensed in reflection of the cavity. The cavity visibility is typically 80 %. The servo loop is designed to have a UGF of more than 100 kHz (in principle, nearly 1 MHz), so that several orders of magnitude of noise suppression can be achieved in the GW band. A typical open-loop transfer function (OLTF) of the reference cavity stabilization servo from O1 is shown in figure 2.5. The main features of the optical (frequency-to-power) transfer function in this loop are the reference cavity pole at 25 kHz, and the pole of the pre-modecleaner36 at 560 kHz (since the frequency actuator is located before the PMC, but the reference cavity is located after it). On the actuation side, we need an actuator that can actuate with minimal phase delay up to 1 MHz, with enough range to cancel the freerunning NPRO noise in and above the GW band. Additionally, it must be able to adjust the laser frequency by several gigahertz on thermal timescales (≲ 1 Hz), in order to compensate for long-term drifts in either the NPRO length or the reference cavity length. To achieve all these requirements, three actuators are employed: 1. on thermal timescales, control is applied to a thermoelectric cooler attached to the NPRO crystal, with a coefficient of order 1 GHz/V; 2. for frequencies between ≃ 1 Hz and ≃ 10 kHz, control is applied to a piezoelectric transducer mounted to the NPRO crystal, with an actuation coefficient of order 1 MHz/V; and 3. for frequencies above ≃ 10 kHz, control is applied to a broadband electro-optic modulator placed at the output of the laser resonator, with an actuation coefficient of order 10 mrad/V. Finally, we note here that the light used to illuminate the reference cavity is frequencymodulated at 79 MHz with an acousto-optic modulator (AOM) placed in a double- Magnitude [Hz/Hz] 22 103 102 101 100 103 104 105 103 104 105 Phase 90◦ 45◦ Frequency [Hz] Figure 2.6: Open-loop transfer function (model and measurement) of the input modecleaner frequency locking loop, with no feedback from the interferometer. pass configuration.h This provides a mechanism for changing the main laser frequency while keeping the light in the reference cavity on resonance: within the bandwidth of the reference cavity servo, adjusting the rf modulation frequency of the AOM produces an offset in the PDH error signal, which the servo cancels by adjusting the frequency of the main laser. Input modecleaner stabilization The input modecleaner consists of three mirrors suspended in vacuum, forming a ring cavity with a 32.9 m round-trip length. The input and output coupling mirrors each have a 0.6 % transmissivity, and the third mirror is a high reflector. The cavity pole is f IMC = 8.8 kHz. The resonance condition between the main laser and the input modecleaner is sensed with PDH reflection locking, using 24 MHz sidebands with a modulation h That is, the beam is sent through the AOM, then the first-order diffracted beam is retroreflected (with a 90 ° polarization rotation) back into the AOM, producing light modulated at twice the AOM’s rf modulation frequency. This cancels pointing fluctuations in the first-order modulated beam induced, for example, by variations in the rf modulation frequency.38 23 index of Γ ≃ 0.01 rad. With 20 W of laser power incident on the modecleaner, the power on the PDH photodiode is PIMC = 50 mW when the modecleaner is unlocked. If the modecleaner is critically coupled (carrier visibility v = 1), the reflected light on the photodiode when locked is entirely from the PDH sidebands, and this sets √ ( ) the shot-noise-limited performance of the loop at 3(hc/λ0 )PIMC Γ2 / 2PIMC Γ/ f IMC = ( )√ f IMC /2 3(hc/λ0 )/PIMC ≃ 15 μHz/Hz1/2 . In practice, the modecleaner is not critically coupled (v < 1) and the reflected light on the photodiode (when locked) is dominated by a few milliwatts of carrier light, giving a shot-noise limited performance √ ( ) ( )√ 2(hc/λ0 )(1 − v)PIMC / 2PIMC / f IMC = f IMC /2Γ 3(hc/λ0 )(1 − v)/PIMC ≃ 0.4 mHz/Hz1/2 . Resonance is maintained by feeding the servo control signal to the AOM in front of the reference cavity, yielding a loop with roughly 50 kHz bandwidth (figure 2.6). When the interferometer is not locked, the low frequency portion of the AOM control signal is relieved by adjusting the modecleaner length, with a crossover frequency of a few tens of hertz. When the interferometer is locked, the modecleaner length is instead adjusted using the PDH error signal from the main interferometer (explained below). Common-mode arm length stabilization Frequency stabilization of the laser to the common-mode arm length is again achieved with PDH locking, using the 9.1 MHz signal in reflection of the power recycling mirror. The optomechanical transfer function from common-mode arm length fluctuation to reflected demodulated power is18 (9I) δPrefl = 4ℵ1 g2p r a′ r sb (2π/c)(ν0 δL + + L + δν) / , 1 + i f f+ (2.21) where ℵ1 = 2J0 (Γ9 )J1 (Γ9 )P0 , g2p is the power recycling gain, r ′a is the derivative of the arm reflectivity for the carrier, r sb is the arm reflectivity for the sidebands, and f + ≃ 0.6 Hz is the common-mode pole (defined in equations 2.19). Based on budgeting of the test masses and the laser, we assume that in the GW band the length fluctuations δL + are negligible compared to the frequency fluctuations δν. (9I) Additionally, δPrefl is sensitive to fluctuations in the PRC length δL p , albeit to a lesser extent than common-mode fluctuations. (9I) On this basis, δPrefl is used as an error signal for frequency-stabilizing the laser to the common-mode arm length. Feedback is achieved by adjusting the modecleaner Magnitude [Hz/Hz] 24 102 101 100 10−1 103 104 105 103 104 105 Phase 90◦ 45◦ 0◦ Frequency [Hz] Figure 2.7: Open-loop transfer function (model and measurement) of the laser frequency locking to the interferometer common-mode arm length. length, which is equivalent to adding an error point offset to the modecleaner frequency servo loop. Because this actuator is mechanical, it is not feasible to achieve a multi-kilohertz actuator this way. Therefore, in addition to the mechanical modecleaner length adjustment, the common-mode error signal is also summed directly into the modecleaner’s electronic error point (another example of “additive offset”). The open-loop transfer function of the common-mode loop is shown in figure 2.7, and the crossover between the mechanical feedback and the additive offset is shown in figure 2.8. This electronic additive offset scheme causes the modecleaner to be slightly detuned from resonance for frequencies above the additive offset crossover, but the bandwidth of the modecleaner is wide enough ( f IMC = 8.8 kHz) that this detuning is negligible. The final piece of the common-mode arm length stabilization scheme is the tidal feedback. The above servo topology uses the common-mode error signal to adjust the modecleaner length, and the resulting modecleaner error signal is used to adjust the rf drive frequency of the reference cavity AOM. As the common-mode arm length drifts relative to the reference cavity length, a low-frequency control signal accumulates on the VCO driving the AOM. This VCO has a control range of ±1 MHz, Magnitude [V/V] 25 101 100 10−1 Phase 10−2 180◦ 135◦ 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ 101 102 103 101 102 103 Frequency [Hz] Figure 2.8: Crossover (model and measurement) from the laser frequency feedback to the modecleaner length feedback. which means that the common-mode arm length can drift by at most ±13 μm before the VCO saturates. On the other hand, daily earth tides can produce several hundreds of microns of displacement over a 4 km basline. To keep the interferometer locked for many hours, tidal compensation is required. To do this, the VCO control signal is offloaded to the upper-intermediate end masses, which each have a dc longitudinal range of 13 μm.39 The bandwidth of this offloading is of order 1 Hz. The low-frequency control that accumulates on each upper-intermediate mass is then offloaded to the hydraulic isolator supporting the suspension,40 with a bandwidth of about 10 mHz. The range of this actuator is about 1 mm. Block diagram A block diagram of the laser frequency stabilization system is given in figure 2.9. The open-loop transfer function of the common-mode arm length stabilization loop is H= ( ) G AK P F/K + M 1 − G AK , (2.22) 26 Hz A(f) V V F(f) V V Hz Hz G(f) T(f) Hz Hz K(f) Hz Hz Hz M(f) P(f) V V Figure 2.9: Block diagram of the Advanced LIGO frequency stabilization system. G( f ) refers to the main laser and the reference cavity stabilization. A( f ) refers to the AOM and its VCO, and servo electronics for feeding the input modecleaner error signal to the VCO. K( f ) refers to the modecleaner and its PDH photodiode. P( f ) refers to the interferometer, its PDH photodiode, and the servo electronics common to the fast and slow feedback paths. M( f ) refers to the slow feedback electronics. F( f ) refers to the fast feedback electronics. T( f ) refers to the tidal feedback path. where G = G/(1 − G) is the closed-loop transfer function of the reference cavity stabilization, and we have neglected the tidal offloading T . Since the UGF of G is above 100 kHz, we have G ≃ −1 below a few tens of kilohertz. Note that since J ≡ G AK (2.23) is the open-loop transfer function of the input modecleaner length loop in the absence of any interferometer feedback, we can write H as ( ) ( ) JP F/K + M H= = JP F/K + M , 1− J (2.24) with J = J/(1 − J). The crossover transfer function from the modecleaner length control to the additive offset control is given by X= 2.3 G AK MP J MP ( ). ( )= 1 − J 1 + FP/K 1 − G AK 1 + FP/K (2.25) Differential arm length control Differential arm length is controlled by sensing the fluctuations in the carrier light at the inteferometer’s dark port and feeding the signal back to one of the end test 27 masses. Since dark port fluctuations are also used to estimate the strain h(t) incident on the detector, the sensing and control of this degree of freedom must be exceptionally low noise, and the transfer function from differential test mass force to dark port power must be well characterized. Table 2.3 lists parameters relevant to the differential arm length control. Optomechanical response The inteferometer’s optomechanical response determines how differential test mass force F (for example, from a GW strain h) produces power fluctuation P at the dark port. While this quantity is fundamentally a force-to-power transfer function P(ω)/F(ω), we often cast it instead as a displacement-to-power transfer function P(ω)/L − (ω), where L − = F/(− M ω2 ) is the differential arm length displacement for free masses each with mass M . In the GW band, deviation from the free-mass assumption can arise if the interferometer is operated with a microscopic phase offset in the SRC length, as such detuned configurations induce an optical spring.41 While the initial LIGO detectors used PDH sensing to read out the dark port signal, the enhanced and advanced detectors instead use a self-homodyne technique called “dc readout”.42 The differential arm length is held with an intentional dc offset (of order 10 pm), so that carrier light from the arms appears at the dark port. When the differential arm length fluctuates (for example, from a GW), it creates carrier audio sidebands that also appear at the dark port and modulate the amplitude of the dark port field. For pure signal extraction, the transfer function from free differential test mass displacement to antisymmetric port fluctuation is18 δPas ( f ) = 2ℵdc g2p g2s r a′ 2 ( ) 2π 2 δL(0) − δL − ( f ) / , λ0 1 + i f f− (2.26) where δL(0) − is the microscopic differential length offset, f − is the optical pole given by (2.20), and ℵdc = 4J0 (Γ9 )2 J0 (Γ45 )2 Pin . (2.27) In any case other than pure signal extraction or pure signal recycling, the optomechanical response involves two additional quantities. The first parameter is the homodyne angle. Homodyne detection techniques such p as dc readout work by beating a signal-modulated field 2E 0 [1 + α(t)] cos[ω0 t + ϕ(t)] p against an unmodulated local oscillator field 2E 0 cos[ω0 t + ζ], where ζ is a static 28 phase offset known as the homodyne angle. For certain homodyne detection topologies (for example, a Mach–Zender interferometer), the homodyne angle may be adjusted by varying the relative path length traveled between the local oscillator field and the signal field (for example, using a piezoelectrically actuated mirror). In ideal dc readout, however, both the local oscillator field and the signal field copropagate inside the interferometer, meaning that the homodyne angle is fixed at ζ = π/2 = 90 °. The addition of a contrast defect produces carrier light in the quadra- ture orthogonal to the local oscillator light, causing a rotation of the homodyne angle. The condition ζ = 90 ° is known to have been satisfied to better than 3 ° during O1, based on the total amount of contrast defect light exiting the dark port with no dc readout light. The second parameter is the SRC detuning. The static, one-way microscopic SRC de(0) (0) tuning phase is ϕ(0) s = (2π/λ0 ) δL s . ϕs = 0 corresponds to pure signal recycling, and ϕ(0) s = π/2 corresponds to pure signal extraction. Other detuning phases result in an optical spring that reduces the displacement-to-power transfer function below the spring frequency; a fuller discussion of this effect is given in chapter 5. Nominally, Advanced LIGO is configured to operate with pure signal extraction for the first few observing runs. However, during O1, H1 operated with about 0.5 ° of unintentional positive (antispring) detuning, likely due to an offset in the angular control of the signal recycling mirror. This was observed as a loss of gain in the optomechanical response around 10 Hz and below, as measured by a calibrated differential arm length radiation pressure actuator (the photon calibrator). The antispring effect was subsequently confirmed by measuring the optomechanical response as a function of (intentional) microscopic SRC length detuning (figure 2.10). Readout The laser light exiting the antisymmetric port contains the dc readout carrier light, the audio-band signal field, the rf control sidebands (mostly the 45.5 MHz sidebands), and carrier light that is not mode-matched into the arm cavities (the “contrast defect” light). The goal of dc readout is to sense the audio-band beat note between the dc carrier light and the carrier audio sidebands (rather than the rf beatnote between the 45.5 MHz light and the carrier audio sidebands, which is the idea of PDH). To do this, an output modecleaner is used to separate the carrier light from the rf sideband light; the carrier light (and its audio sidebands) are then directed onto 29 Magnitude [mW/pm] 4. 0 3. 5 3. 0 2. 5 2. 0 1. 5 1. 0 0. 5 0. 0 101 102 103 Phase 0◦ −45◦ +0.3 nm = +0.1◦ +2.3 nm = +0.8◦ −90◦ +4.3 nm = +1.5◦ 101 102 Frequency [Hz] 103 Figure 2.10: Optomechanical response of the interferometer’s differential arm length, shown for several different SRC detunings at 10 W of input power. Solid lines show estimated theory curves. Fundamentally, this transfer function is a force-to-power transfer function P( f )/F( f ), but here (as is customary) the force is referred to the displacement x = −F/M ω2 of a free mass. two InGaAs photodiodes, each with a 3.0 mm diameter. Each photodiode is reversebiased with +12 V.49 In vacuum, the photocurrent from each photodiode is shunted across a resistor, and the resulting voltage is read out with an LT1128 op-amp. The shunt resistor is selectable, and may be either 100 Ω or 400 Ω.50 The summed photocurrent of these two photodiodes is the differential arm length error signal. Two photodiodes (rather than one) allow for a greater amount of dc offset light to be used, and also provide an important diagnostic tool (the “null stream”), formed by differencing the two photocurrents. The output modecleaner is kept resonant by adjusting one of its two piezoelectri- 30 Photodiode current (total) 20 mA dc Photodiode quantum efficiency 0.88 Photodiode responsivity 0.75 A/W 43 4.2(2) mW/pm Optical gain Optical pole43 340(20) Hz Microscopic arm detuning (L x − L y ) 12.5(4) pm Microscopic RSE detuning (one-way)44 1.5(6) nm 45 90(3) ° i;46 427 μs Homodyne angle Time delay Freerunning noise47 Residual noise 47 0.6 μm rms 60 fm rms Actuator EY Loop bandwidth 45 Hz Relief bandwidth to penultimate mass 20 Hz Relief bandwidth to antipenultimate mass 2 Hz Relief bandwidth to hydraulic actuator i 11 mHz Includes optical time delay (L/c = 13 μs),37 digital delays,48 and the delay from uncompensated high-frequency transfer function features.46 Table 2.3: Parameters for differential arm length sensing and control for aLIGO H1 during the first observing run. The parameters for L1 are similar, although no RSE detuning was observed. cally actuated mirrors. The resonance condition is sensed by applying a 4.1 kHz dither to this mirror and demodulating the resulting line in the GW signal. Control The differential arm length control signal is fed back to one test mass only. This also alters the interferometer’s common-mode arm length, but the frequency stabilization servo strongly suppresses these fluctuations. The UGF of the differential arm length loop is about 50 Hz. The OLTF is shown in figure 2.11. Three out of the four stages of the suspension are used to displace the test mass; the crossover transfer functions for this control hierarchy are shown in figure 2.12. −90◦ −45◦ 0◦ 45◦ 90◦ 135◦ 180◦ 10−1 10−1 100 100 102 102 Frequency [Hz] 101 101 103 103 −180◦ −135◦ −90◦ −45◦ 0◦ 45◦ 90◦ 135◦ 180◦ 10−3 10−2 10−1 100 101 102 101 101 Frequency [Hz] 102 102 103 103 version of the left-hand plot, with the measurement shown.51 Figure 2.11: Open-loop transfer function model of the differential arm length control. The right-hand plot is a zoomed-in −180◦ −135◦ Magnitude [m/m] Phase Magnitude [m/m] Phase 108 107 106 105 104 103 102 101 100 10−1 10−2 10−3 10−4 10−5 31 32 Magnitude [m/m] 101 100 10−1 Penultimate Upper intermediate 100 101 102 100 101 102 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 2.12: Crossover transfer functions for the penultimate and upperintermediate actuator offloading for the differential arm length control. Since dc readout is intrinsically quadratic, differential arm length motion at a frequency f can produce upconverted fluctuation in the readout at a frequency 2 f . To keep this upconverted noise well below the true differential arm length noise, the allowed residual differential arm length fluctuation is set at 1 fm.35 During the first observing run, the residual fluctuation was instead a few tens of femtometers.47 However, dc readout upconversion could not be shown to limit the interferometer noise performance in the GW band.52 2.4 Noise from vertex length controls 33 PRC Michelson SRC Photodiode power 17 mW dc 17 mW dc 17 mW dc Optical gain 3.6 W/μm 0.60 W/μm 0.13 W/μm Freerunning noise 0.9 μm rms 0.4 μm rms 0.1 μm rms Residual noise 1.0 pm rms 5 pm rms 20 pm rms Beamsplitter SRM Actuator PRM Loop bandwidth 65 Hz 10 Hz 25 Hz 60 mHz 30 mHz 60 mHz Upper mass relief bandwidth Table 2.4: Parameters for vertex length sensing and control for aLIGO H1 during the first observing run. The parameters for L1 are similar. H1 vertex length sensing, 2016–01–13 £ ¤ POP9 W/µm £ ¤ POP45 W/µm Q Q 2. 0 4 1. 5 3 1. 0 2 0. 5 1 −I I −Q −I PRM Beamsplitter SRM I −Q Figure 2.13: Sensing matrix of the vertex degrees of freedom. The cross-coupling of the beamsplitter from a Q (pure Michelson) signal into I (PRC or SRC) signal can be seen in the 45 MHz channel. In the 9 MHz measurement, the PRC and beamsplitter elements are on top of each other. Optical response and readout Per watt of incident photodiode power, the vertex degrees of freedom are best sensed at the power-recycling pick-off port (POP), rather than the reflected port or the antisymmetric port. Using POP to sense the vertex lengths requires very strong suppression of common-mode fluctuations, as this signal shows up more strongly 34 in POP9I and POP45I than any of the vertex lengths. Once this requirement is satisfied, the next strongest signal is the PRC length, which shows up most strongly in POP9I. The Michelson length shows up most strongly in POP45Q. Finally, the SRC length shows up most strongly in POP45I. However, the PRC length also shows up in POP45I, with greater strength than the SRC length.18 Therefore, the error signal for the SRC length control uses a combination of POP45I and POP9I which is insensitive to PRC length fluctuation. The sensing matrix of the POP sensors is shown in the radar plots in figure 2.13. There remains the question of how to choose the UGFs of these three loops. First, PRC length fluctuations show up more strongly than SRC length fluctuations in both POP9 and POP45; therefore, the PRC length UGF should be higher than the SRC length UGF. Second, if the beamsplitter is used to control the Michelson length, there will be cross-coupling of Michelson actuation into power- and signal-recycling lengths; therefore, the UGF of the Michelson loop should be lower than either the PRC and SRC loops. These two considerations set the Michelson loop as the low- est UGF loop, and the PRC loop as the highest UGF loop. Table 2.4 gives relevant parameters for these loops. The residual rms requirement for the vertex length loops is set at 1 pm, based on the linearity limit of the photodetector.35 During the first observing run, the actual residual rms of the loops was less than 20 pm (figures 3.8, 3.9, and 3.10). The OLTFs of the vertex loops are shown in figure 2.14. Cross-coupling into differential arm length Optomechanical cross-coupling The differential arm length error signal e − has contributions not only from freerunning displacement noise (L − ) and photodiode sensing noise ( n − ), but also from residual noises from other degrees of freedom—particularly the Michelson length and the SRC length. The Michelson coupling into the differential arm length readout is a straightforward phase coupling: Michelson motion produces differential phase sidebands, which are converted to amplitude sidebands at the antisymmetric port in the same manner as differential phase sidebands from the arms. Because the length-to-phase conversion for differential arm length fluctuations is enhanced by g2a = 2F/π ≃ 280 −90◦ −45◦ 0◦ 45◦ 90◦ 135◦ 180 ◦ 10−1 10−1 101 101 Frequency [Hz] 100 100 102 102 SRC 103 103 Michelson PRC −180◦ −135◦ −90◦ −45◦ 0◦ 45◦ 90◦ 135◦ 180 ◦ 10−3 10−2 10−1 100 101 102 101 Frequency [Hz] 102 SRC 101 103 103 Michelson PRC 102 results of measurements; lines show the corresponding models. The right-hand plots are zoomed in from the left-hand plots. Figure 2.14: OLTFs of the vertex length degrees of freedom, shown across two different frequency ranges. Points show the −180◦ −135◦ Magnitude [m/m] Phase Magnitude [m/m] Phase 108 107 106 105 104 103 102 101 100 10−1 10−2 10−3 35 36 ns rs xs Ps ks Ks rs es optomechanical coupling feedforward filter Ξ F ks n− r− x− P− k− K− e− Figure 2.15: Illustration of optomechanical coupling and feedforward subtraction, here for the SRC length loop coupling into differential arm length. Residual SRC length fluctuation r s , which comprises both suppressed displacement noise xs /(1 − Hs ) and impressed sensing noise Hs n s /(1 − Hs ), couples into the differential arm length readout via some optomechanical coupling Ξ. To reduce the influence of this coupling on the differential arm length readout, the SRC length control signal k s is summed (either mechanically, digitally, or analogly) into the differential arm length error point via some feedforward filter F . In general, it is not possible to construct a filter F which simultaneously cancels the displacement and sensing noise contributions to r s . relative to the length-to-phase conversion for Michelson fluctuations, the expected coupling of Michelson motion into differential arm length readout (expressed as an equivalent differential arm length motion) is27 Ξm = 1 g2a = π 2F ≃ 3.6 × 10−3 m/m. The measured coupling is shown in figure 2.17. (2.28) 37 Magnitude [m/m] Without feedforward 10 −3 10−4 10−5 10−6 101 102 103 101 102 103 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 2.16: Coupling of power recycling mirror motion into differential arm length readout. The signal recycling length coupling into the differential arm length readout has several parts. The first is a 1/ f 2 coupling mediated by radiation pressure: SRC phase sidebands propagate into the arms and produce differential intensity fluctuations, which are converted to differential length fluctuations via radiation pressure.18,27 The second is a high-frequency coupling rising like f 2 ; it appears if the SRC is microscopically detuned from pure resonant sideband extraction.27 These two effects together imply a coupling of SRC motion into differential arm length motion:27 ( ) δL(0) Pa F 10 Hz 2 − Ξs = 0.012 m/m × × × × 750 kW 10 pm 450 f ( )2 (0) (0) δL − F f ϕs −5 × × × + 3 × 10 m/m × . 10 ° 10 pm 450 100 Hz (2.29) Magnitude [m/m] 38 Without feedforward With feedforward 10−2 10−3 10−4 101 102 103 101 102 103 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 2.17: Coupling of beamsplitter motion into differential arm length readout. Finally, another class of high-frequency couplings, generally rising like f or faster, can arise from nonidealities such as arm imbalances and angular control offsets,53 but these effects are not easily quantified analytically. The measured SRC length coupling into the differential arm length readout is shown in figure 2.18. Additionally, the measured PRC length coupling into the differential arm length readout is shown in figure 2.16. Feedforward subtraction The cross-couplings described above seem to set a hard limit on the allowed residual fluctuation of the vertex degrees of freedom, based only on the desired differential arm length sensitivity and the magnitude of the cross-coupling. However, because the residual vertex fluctuations are witnessed by other sensors, it is possible to sub- 39 Magnitude [m/m] 10−2 Without feedforward With feedforward (control) With feedforward (displacement) 10−3 10−4 10−5 10−6 101 102 103 101 102 103 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 2.18: Coupling of signal recycling mirror motion into differential arm length readout. Note in this case that because the bandwidth of the SRC loop is low, the application of feedforward subtraction does not worsen the displacement noise coupling into the differential arm length readout. tract off these cross-coupled signals from the differential arm length.54 This may be done after the fact—using stored sensor data—or it may be done mechanically in real time by feeding the expected cross-coupled signals forward onto the test masses. This latter strategy was employed for the first observing run. Figure 2.15 shows an implementation of feedforward subtraction of SRC length into differential arm length. The residual SRC length noise is rs = xs + K s n s , 1 − Hs (2.30) where xs is the loop’s freerunning displacement noise, and n s is the loop’s sensing noise. The term K s n s /(1 − Hs ) is therefore the “control noise” impressed onto the SRC length by the sensor. 40 An optomechanical coupling Ξ( f ) produces an apparent noise Ξ × r s in the differential arm length readout. To cancel this coupling, we can design a filter F( f ), apply it to the SRC length control signal k s , and apply the product F × k s to the differential arm length actuation to cancel Ξ × r s . However, since ks = P s xs + K s n s ̸= r s , 1 − Hs (2.31) it is in general not possible to design a filter F which simultaneously cancels the couplings from both xs and n s . If we wish to cancel the coupling from xs , we should choose F = Ξ/Ps . On the other hand, if we wish to cancel the coupling from n s , we should choose F = Ξ. Note that feedforward that is tuned to cancel the coupling of xs may worsen the coupling of n s , and vice versa. For the first observing run, feedforward was employed for the Michelson and SRC lengths by actuating on the penultimate stages of the input test masses. The feedforward was tuned so as to cancel sensing noise, since this noise was expected to dominate over the intrinsic cavity noise. However, because the Michelson and SRC loop bandwidths are relatively low, the feedforward filters are able to cancel sensing noise without appreciably worsening the displacement noise couplings. 2.5 Angular control This section presents an overview of the angular sensing and control for the Hanford interferometer. Table 2.5 enumerates the angular loops and gives their bandwidths. Readout techniques Two general classes of interferometric angular readout are employed in Advanced LIGO: wavefront sensing, which is an extension of the PDH technique, and audio- band pointing. Both types of loop operate on the principle that misalignment of a TEM00 beam relative to an optical cavity (and vice versa) generates TEM01 light. Beat notes involving TEM01 light against TEM00 light are sensed using quadrant photodetectors, with the photocurrents differenced as appropriate to generate either pitch or yaw signals. In the case of wavefront sensing, the signal of interest is generated from the beat of TEM00 audio-band light against TEM00 rf sideband light, or vice 41 Loop Sensor Bandwidth [Hz] Pitch Yaw Diff. hard AS 45 MHz 2.1 ♢ 3.6 ♢ Comm. hard Refl. 9 MHz <0.1 <0.1 Diff. soft Arm trans. dc <0.05 <0.05 Comm. soft Arm trans. dc <0.05 <0.05 PRM POP dc 0.03 0.2 PR2 — — — PR3 Refl. 9+45 MHz 0.12 ♢ 0.08 Beamsplitter AS 36 MHz 2 SRM AS 36 MHz 0.13 0.4 SR2 AS dc 1.2 ♢ 2 SR3 Shadow sensor ∼0.03 — ITM Optical lever Input pointing Refl. 9 MHz 3 ♢ ♢ — 0.10 0.06 Table 2.5: Sensors for and bandwidths of the angular control loops for H1. Values marked by ♢ were determined by OLTF measurement, and refer to the highest UGF where applicable. All others were determined by step response. No yaw control was implemented for SR3, and no control was implemented for PR2. versa.j;55,56 In the case of audio-band pointing, the signal of interest is generated by TEM01 audio sideband light against TEM00 carrier light, or vice versa. Beam misalignment may take the form of a tilt, a translation, or both, and hence fully sensing the misalignment requires at least two quadrant detectors, separated by some amount of Gouy phase (preferably 90°). Arm loops Optomechanical plants The eight arm loops constitute the most phenomenologically rich angular control loops. The power stored in the arms generates not only radiation pressure that affects the interferometer’s length responses, but also radiation torques that affect j That is, TEM 01 rf light against TEM00 carrier light. 42 hard soft Figure 2.19: Diagram of hard and soft cavity optic motions. The hard mode is so named because the radiation torque enhances the mechanical stiffness of the optic suspensions, while the soft mode diminishes it. the interferometer’s angular responses. The effect of these radiation torques was first analyzed by Sidles and Sigg,57 and we summarize the relevant points here. In the absence of radiation pressure, the transfer function taking test mass torque to test mass angle is determined only by the mechanics of the mass and its suspension; for Advanced LIGO, the relevant torsional constants are κp = 9.7 N m/rad and √ κy = 9.4 N m/rad.58 The corresponding angular frequencies are f p,y = (1/2π) κp,y /I p,y . In the presence of radiation pressure, the four test masses are coupled together, and the eigenmodes of the coupled system involve simultaneous motions of the test masses together. The eigenvalues of the system give two optical torsional constants:58 √ LP (g e + g i ) ± (g e − g i )2 + 4 κs,h = × , c ge gi − 1 (2.32) where g i,e = 1 − L/Ri,e is the g factor of each test mass. These torsional constants are added to the mechanical torsional constant of the suspension in order to give the overall torsional constant of the mode. The hard and soft modes of a two-mirror cavity are illustrated in figure 2.19. An important consequence of this particular eigenvalue problem is that one of these eigenvalues (κh ) is always positive, and the other is always negative (κs ). Therefore, one set of modes has a torsional constant κ = κp,y + κh that is stiffer than the Magnitude [rad/rad] 43 Pitch Yaw 101 100 10−1 100 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ 100 Frequency [Hz] Figure 2.20: Open-loop transfer functions for angular control of the differential hard mode. The multiple UGFs arise from imbalances between the test mass actuators, overly aggressive test-mass bounce-mode cutoff filters, and insufficient boosting in the digital compensator. These issues were resolved after O1. mechanical torsion (the “hard modes”), and the other set has a torsional constant κ = κp,y +κs that is softer than the mechanical torsion (the “soft modes”). If |κs | > κp,y , the soft modes become unstable, and require active control with a bandwidth sufficient to stabilize the unstable poles. In Advanced LIGO, the test mass g factors ( g i = −0.78 and g e = −1.06) are chosen in part to make |κs | small: with 800 kW of power in each arm, the primary hard mode resonance frequency is expected to be around 3 Hz, and the soft mode resonance is expected to be around −0.3 Hz. 44 Sensing and actuation Based on simulations by Barsotti et al.58 and some experimental trial and error, the following sensors were chosen for the arm modes: • the 45 MHz antisymmetric port WFSs are used to sense the differential hard mode (OLTF shown in figure 2.20); • the 9 MHz WFSs in reflection are used to sense the common hard mode; and • the end station dc QPDs are used to sense the common and differential soft modes. For the WFS loops, the general procedure for producing a suitable error signal is given in the following. 1. Ensure that a suitable centering loop is closed around each WFS. 2. Drive a line in the relevant DOF. Verify that the line appears in every quadrant of the relevant WFSs. 3. Rotate the (digital) demodulation phases so that the signal shows up in the same quadrature for each quadrant. Verify that the line appears with roughly the same amplitude in each quadrant. 4. Choose a linear combination of the WFSs that gives a usable error signal. “Usable” here is somewhat ill-defined, but we can identify some desirable characteristics. a) The error signal crosses zero in the appropriate place; for example, in a place where some power buildup is maximized. If each individual WFS signal has an undesired dc offset, an appropriate combination of two signals may sometimes be found with a zero crossing in the right place. b) The error signal is relatively insensitive to other DOF(s). If each individual WFS senses some DOF much more strongly than the DOF of interest, an appropriate combination of two signals may sometimes be found which is sensitive to the DOF of interest but insensitive to the other DOF. Rejection of other DOFs in the error signal may be achieved after the fact by diagonalizing the interferometer’s angular sensing matrix. For the hard loops, the error signals were initially chosen to give good zero crossings (no suppression of other DOFs was considered). For the differential hard loops, a relatively high bandwidth (more than 3 Hz) is required to keep the antisymmetric port power stable enough to keep the interferometer locked with zero commonmode arm length offset. The common-mode hard loops may also be operated at 45 similarly high bandwidths. However, during O1, the coupling of the sensor noise from the 9 MHz reflection WFSs into the differential arm length readout was so high that these loops were operated with lower bandwidth. For the pointing loops, the procedure signal selection procedure is similar, although no phasing is required. For the soft loops, the QPD combination at each end station was chosen to minimize the sensitivity of the signal to motion of the suspension which supports the QPDs.59 Importantly, this does not minimize the sensitivity of the resulting error signals to hard motion, and indeed hard motion is seen with good SNR in the soft loop error signals. Therefore, the stable operation of these loops relies on gain hierarchy, with the soft loop UGFs kept well below the hard loop UGFs. The positions of the end station spots were chosen to maximize the interferometer recycling gain. For all the arm loops, the penultimate test masses are used as actuators. Importantly, in order for the loops to be properly coupled into the hard/soft basis, the test mass actuation transfer functions must be made uniform; this requires that top-stage local damping is uniform, and that the penultimate stage actuators are balanced. The relatively sharp zeros observed in 2.20 are indicative of actuator mismatch between test masses. In addition to the interferometric loops listed here, local pitch damping (via optical levers) was also applied to the input test masses during O1 in order to avoid the common-mode soft instability (see below). This also contributes to the actuator mismatch for the hard and soft loops. Symmetric port loops On the symmetric port side, six degrees of freedom are controlled. 1. The difference of the 9 MHz reflection WFSs is used to sense the input pointing of the beam into the interferometer. The actuator is the final steering mirror into the interferometer (IM4). 2. During O1, a combination of the 45 MHz reflection WFSs is used as an error signal to control the angle of the large PRC folding mirror (PR3). The 9 MHz WFS were also used to subtract common-mode hard motion from the PR3 error signal. 3. The pointing of the PRC light onto one of the PRC QPDs is used to control the angle of the power recycling mirror. Magnitude [rad/rad] 46 Pitch Yaw 101 100 10−1 100 101 100 101 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 2.21: Open-loop transfer functions for angular control of the beamsplitter. For both H1 and L1, the 45 MHz reflection error signal suffered from the problem that the optical gain would flip sign as the interferometer’s recycling gain crossed a certain threshold (somewhere in the region from 30 to 33 W/W). For H1, this so-called “Arai singularity” was avoided in practice by adjusting the green initial alignment references so that the interferometer would have a high recycling gain (more than 36 W/W) as soon as arm resonance was achieved. To get rid of the Arai singularity for good, both L1 and H1 installed an in-air WFS to monitor the PRC light, with L1 using the 36 MHz signal and H1 using the 45 MHz signal. Beamsplitter and antisymmetric port loops Four degrees of freedom are controlled: 1. A combination of 36 MHz antisymmetric port wavefront sensors are used to control the beamsplitter angles (OLTFs shown in figure 2.21). Magnitude [rad/rad] 47 Pitch Yaw 101 100 10−1 100 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ 100 Frequency [Hz] Figure 2.22: Open-loop transfer functions for angular control of the signal recycling cavity pointing. 2. A combination of 36 MHz antisymmetric port wavefront sensors are used to control the angles of the signal recycling mirror. 3. A QPD at the output of the SRC is used to control the angles of the small SRC folding mirror (SR2) (OLTFs shown in figure 2.22). 4. A shadow sensor is used to control the pitch of the large SRC folding mirror (SR3). The antisymmetric port 36 MHz wavefront sensors presented little difficulty for L1, but were extremely problematic for H1, and their (mis)behavior is part of the constellation of issues surrounding the Hanford signal recycling cavity. Phasing the 36 MHz sensors was difficult: certain quadrants showed little to no response to optical signals, or sometimes showed different responses depending on whether an optic was driven in pitch or yaw. Loops could still be closed around the beamsplitter 48 and SRM angles, but the error signals at 2 W were unusable at 25 W. For O1, stable 25 W operation was achieved by choosing a combination of 36 MHz signals that kept the rf sideband powers high in the recycling cavities. Input modecleaner Angular control of the input modecleaner is achieved with a combination of rf wavefront sensing and dc pointing sensing. Two wavefront sensors are placed in reflection of the modecleaner, and these are used to control (relative to the input beam) the angle of the high reflector and the common-mode angle of the input/output coupling mirrors. The third degree of freedom of the mirrors (the differential angle of the input and output coupling mirrors) is left uncontrolled. Finally, the pointing of the beam on the high reflector is controlled by closing a loop around a dc quadrant photodetector placed in transmission of the high reflector. Output modecleaner To keep the output modecleaner aligned with respect to the antisymmetric port beam, four dither lines are applied to the first and third (out of three) antisymmetricport steering mirrors that steer the beam from the output Faraday isolator into the output modecleaner. These lines are demodulated in the GW signal and used as error signals to drive the third steering mirror and the modecleaner suspension.60,61 For future observing runs, this dither alignment scheme may be replaced by a test mass drumhead sensing scheme, as was employed in Enhanced LIGO.26 Cross-coupling into differential arm length Several effects cause angular control signals to produce differential arm length motion via simple linear coupling. First, if the beam spot is miscentered on the test mass by an amount δ, angular motion θ of the mass will produce a length coupling θδ. Second, imbalance of the suspension actuators can convert an angular drive signal into a partially longitudinal drive signal. Finally, when driving the penultimate stage in pitch, there is an intrinsic mechanical coupling to test mass length. To reduce this linear angular coupling, frequency-independent feedforward is employed on each test mass. A line in pitch or yaw is injected into the relevant penultimate stage angular control signal, and this line then appears in the differential 49 arm length readout because of the linear angular cross-coupling. The angular control signal is then summed into the penultimate stage longitudinal control signal, with some coefficient that is optimized to minimize the appearance of the line in the differential arm length. More generally, the differential arm length noise due to angular fluctuations is the frequency-domain convolution of the motion of the interferometer mirrors.58,62 This means that the angle-to-length mechanism is bilinear, and therefore cannot be subtracted off with linear feedforward. When the dominant angular noise contribution to the differential arm length is no longer linear, the best strategy may be to reduce the mirror motion, for example, by improving the seismic isolation of the mirrors or decreasing the angle-referred shot noise of the sensors. 2.6 Suspension control Test mass suspensions Each test mass suspension comprises two quadruple suspensions, mounted from an active seismic isolation platform40 inside the vacuum system. One of these suspensions contains the test mass as its final payload; it serves to isolate the test ( ) mass from external vibrations, with a transfer function magnitude of 10−15 m/m × ( / )8 100 Hz f in the GW band. The upper three masses are fitted with permanent magnets so that they may be magnetically actuated. The other suspension contains masses fitted with sensors and actuators for controlling the main suspension. On the upper three masses, the actuators are solenoids that act on the permanent magnets of the main suspension masses, and the sensors are shadow sensors. On the bottom mass, the actuator comprises a set of electrodes which actuate capacitively on the test mass itself. The choice of electrostatic—rather than magnetic— actuation is driven by the desire to avoid direct coupling of ambient magnetic fields to the test mass, and to avoid spoiling the Q of the test mass by attaching magnets onto it. The angle of each test mass can be sensed with an optical lever. A diagram of the test mass and the associated suspensions is shown in figure 2.23. Test mass and electrostatic actuator To actuate on the test mass, the outer annulus of the face of the glass reaction mass is patterned with five interdigitated gold electrodes. Four of these electrodes 50 top upper intermediate penultimate test Figure 2.23: Diagram of the Advanced LIGO quadruple suspension. The test mass and penultimate mass are made from fused silica, and are attached with four fused silica fibers. The reaction mass is also made from fused silica. The remainder of the masses and wires are steel. are each confined to a specific quadrant of the reaction mass; the fifth extends throughout the patterned area. Applying differential voltage between any of these electrodes produces a fringing field E inside the test mass, which induces a polar[∫ ] ization field P = ε0 χE, and hence a force F = ∇ d3 r P · E . Since F is proportional to E 2 , this actuator is intrinsically nonlinear. To achieve linear actuation, a fixed bias voltage Vb is applied to the fifth, extended electrode. Then applying a signal voltage v to any of the other electrodes will produce a corresponding force proportional to (Vb + v)2 ≃ Vb2 + 2Vb v. For longitudinal actuation on the end test masses, the proportionality constant is α ≃ 1.5 × 10−10 N/V2 . This is set by the geometry of the fringing field with respect to the test mass, as well as the presence of charge on the mass.16 For the input test masses, the proportionality constant is weaker, because the gap between the reaction mass and the test mass 51 is larger for the input suspensions (20 mm versus 5 mm) in order to reduce squeeze film damping. During O1, the end Y electrostatic actuator was used to control the differential arm length, with 380 V of bias, and a signal range of ±20 V on each quadrant. The electrostatic actuation strength is affected by the presence of charge on and around the test mass.16 Penultimate mass The penultimate mass is fabricated from fused silica, and is fitted with four magnets. On all four suspensions, this stage is used as the main actuator for the test mass angular control. On the end Y suspension, this stage is used for differential arm length feedback (along with the upper intermediate and test stages). On the input X and input Y suspensions, this stage is used for feedforward cancellation of Michelson and SRC length control. Additionally, this stage is used to damp the eight silica violin modes on each test mass. The first harmonics of these modes have frequencies close to f v = 500 Hz. These modes become problematic if their rms amplitude produces more than 10−14 m of differential arm length motion, as this will saturate the dc readout photodiodes when operated in high-sensitivity mode. Worse still, violin modes exhibit dilution damping,63 with Q > 108 . This implies a time constant τ = Q/π f v > 17 h, so it is imperative that they be actively damped if they ring up. Violin modes are sensed in the differential arm length readout, which is bandpassed to produce various control signals. These signals are sent to the penultimate stage, with the option of driving in some combination of length, pitch, or yaw. Upper intermediate mass The upper intermediate mass is made from steel, and is fitted with four magnets, giving longitudinal, pitch, and yaw control. During the first observing run, the end Y upper intermediate mass was used to offload differential arm length control from the test mass. Additionally, both the end X and end Y upper intermediate masses were used to offload the VCO control signal for the reference cavity AOM. The input X and Y upper intermediate masses were not used for H1. 52 Top mass The top mass is made from steel, and is fitted with six magnets. A key feature of the suspension design is that all degrees of freedom of the pendulum masses are supposed to be observable using the top mass sensors. This allows six ac-coupled servo loops to be closed around the top mass in order to damp these degrees of freedom from roughly 0.1 to 1 Hz. The modes are considered well damped if a step response applied to each top mass degree of freedom produces a response with a Q of a few. In the lead-up to O1, it was found that the test mass bounce and roll modes could not adequately be damped using the top-mass sensors alone; the modes would ring up and saturate the differential arm length readout. The bulk of the energy in these modes is stored in the silica fibers between the penultimate mass and the test mass, so these modes are long-lived, with a Q of the same order as the intrinsic silica Q (more than 105 ). Generally, these modes are considered problematic if their amplitude in the differential arm length readout exceeds 10−13 mrms . The coupling of the bounce mode is thought to be well understood: if the local gravity at the test mass does not lie in the plane normal to the beamline, then bounce mode motion produces a small amount of beamline motion. For a perfectly spherical earth with a homogeneous local gravity field, the expected bounce coupling is (2.0 km/6400 km) = 3 × 10−4 m/m; in reality, gravimetry measurements imply bounce couplings ranging from 8 × 10−6 m/m to 6.4 × 10−4 m/m.64 The coupling of the roll mode occurs when the plane of the mode rotation is not parallel to the test mass face; this could arise, for example, from the wedge of the test mass substrate.65 During O1, the four bounce modes were sensed by bandpassing the differential arm length readout, and feeding the resulting to the vertical actuators on the top mass with an experimentally determined phase delay. The four roll modes were sensed by bandpassing the 45 MHz antisymmetric-port wavefront sensor signals and feeding the result to the top mass vertical actuators. Additionally, the top mass receives low-frequency pitch and yaw control signals from the penultimate stage. 2.7 Intensity control Several orders of magnitude of intensity noise suppression are required for Advanced LIGO to achieve its designed sensitivity. The injection-locked amplifier pro- 53 ) ( / ) duces a typical RIN of less than 10−5 /Hz1/2 × 100 Hz f for f < 1 kHz. The re( quirement for the light at the interferometer input is 10−7 /Hz1/2 at 100 Hz and 2 × 10−9 /Hz1/2 at 10 Hz.35 Two intensity stabilization loops are implemented to achieve this suppression. A third loop is available to suppress optomechanical instabilities around the test mass suspension resonances. Inner loop For the inner loop, a pair of photodiodes (one in-loop and one out-of-loop) is used to sense the light exiting one of the high-reflector ports of the pre-modecleaner. The in-loop signal is filtered and then used to actuate on a single-pass AOM located immediately before the pre-modecleaner (figure 2.4). The bandwidth of this loop is approximately 50 kHz, and the shot-noise-limited performance of the loop is sufficient to achieve a RIN of a few parts in 10−8 /Hz1/2 from 30 Hz to 1 kHz.36 However, after the high-power oscillator was activated in H1, the inner loop ceased to properly stabilize the laser intensity, leading to 40 times more RIN entering the interferometer at 100 Hz.66,67 This behavior could be explained by the inner loop diodes seeing a common intensity noise (for example, from jitter or polarization) that is not seen on the other two transmission ports of the pre-modecleaner. Outer loop Additional intensity noise suppression is achieved with an outer loop located in transmission of the input modecleaner. This loop uses an array of four in-loop and four out-of-loop photodiodes, and the loop bandwidth is about 10 kHz. In principle, at 10 Hz the loop can achieve a RIN of 2 × 10−9 /Hz1/2 . Additionally, the interferometer passively filters the intensity noise above the common-mode cavity pole, f + ≃ 0.6 Hz. Outermost loop For powers above 10 W in H1, an optomechanical instability was sometimes observed, producing test mass pitch motion in the common-mode soft degree of freedom at a frequency close to the primary suspension resonances (around 0.5 Hz). This instability could not be cured by adjusting spot positions on the test masses. To reduce the frequency of occurrence, optical lever damping was applied to the 54 input test masses. While generally sufficient for O1, this did not resolve the issue, and the instability again started to appear as commissioning progressed to more than 30 W. Driggers et al.68 eventually determined that this instability could be explained by a nonzero coupling from test mass angle into arm power (of unknown origin), with a magnitude ∂P/ ∂θ ∼ 1010 W/rad at 0.5 Hz. The arm power fluctuation drives the test masses longitudinally, which also drives the test mass angles (in a common-mode soft configuration) because of mechanical length-to-pitch coupling in the test mass suspensions, with an expected magnitude ∂θ/ ∂F ≃ 4 × 10−2 rad/N at 0.5 Hz. Therefore, common-mode soft test mass motion propagates around an optomechanical feedback loop, with an open-loop gain (2/c)( ∂P/ ∂θ)( ∂θ/ ∂F) that can exceed unity at 0.5 Hz. To suppress this instability, a third intensity stabilization loop was created in order to stabilize the power in the arms around the suspension resonance frequencies. The transmitted powers from each end station (measured by the in-vacuum QPDs) are summed, giving an error signal that measures the average transmitted arm power. The digitally filtered error signal is summed electronically into the outer loop, giving an open-loop gain that is above unity between 0.2 and 0.7 Hz.69,70 55 3 Noise in the Advanced LIGO detectors This chapter presents a budget of the noise in the Advanced LIGO Hanford detector. Noise analysis of the Livingston detector is given by Martynov16 and Martynov et al.43 Similar budgets for the initial detectors have been given, for example, by Adhikari71 and Ballmer.72 As of this writing, the budgets for both Advanced LIGO detectors fail to explain the differential arm length sensitivity between 50 Hz and 200 Hz; the source of this noise is under active investigation. 3.1 Quantum noise Overview Quantum noise refers to the joint contributions of radiation pressure noise (a displacement noise) and shot noise (a sensing noise) to the differential arm length readout. Quantum noise has been explained by Caves73 as the effect of vacuum fluctuations which enter the unused ports of the interferometer—particularly the antisymmetric port. In the case of no signal recycling mirror, the quantum noise of a lossless interferometer is41,74 ( ) 2 1 h SQL S hh ( f ) = K + , K 2 where h2SQL = h π3 ML2 f 2 and K= Pb c π3 λ0 ML2 f a2 f 2 × 1 . 1 + ( f / f a )2 (3.1) (3.2) £ ¤ ASD of displacement m/Hz1/2 56 Uncorrelated noise („null“) Total noise („sum“) Photodiode cross-correlation Expected coating thermal noise 10−19 10−20 102 103 Frequency [Hz] Figure 3.1: GW photodiode auto- and cross-spectral densities from the first observing run, referred to freerunning differential arm length. The photodiode sum is the usual freerunning DARM estimate. The photodiode null shows only uncorrelated noises, such as shot noise and photodiode electronics noise. The cross-correlation shows only correlated noises, such as displacement noise and sensing noises arising in the interferometer. The effect of adding a signal recycling mirror with resonant sideband extraction (that is, π/2-detuned signal recycling) is to make the substitutions fa → y fa and Pb → Pb /y, (3.3) with y = (1 + r s )/(1 − r s ). The effect of resonant sideband extraction, then, is to decrease the effective power on the beamsplitter and broaden the linewidth of the arms. Photodiode cross-correlation The GW readout is sensed with two photodiodes (called “A” and “B”), each receiving half the light transmitted through the output modecleaner. The shot noise on these photodiodes is uncorrelated. Therefore, the cross-spectral density S AB ( f ) of the two photocurrents i A and i B can be used to estimate the amount of correlated noise i 0 present in the interferometer or the GW readout, since the shot noise will average away. An example is provided in figure 3.1, which shows the correlated differential arm length noise during the first observing run. 57 How many averages are required to resolve the correlated noise? The estimated coherence γ̂2AB,(N) after N averages is γ̂2AB,(N) ( f ) = = ¯ ¯ ∗ ¯〈 i i B 〉 N ¯2 A 〈 i ∗A i A 〉 N 〈 i ∗B i B 〉 N ¯ ∗ ¯ ¯〈 i 0 i 0 〉 N + 〈 i ′ ∗ i 0 〉 N + 〈 i 0∗ i ′ 〉 N + 〈 i ′ ∗ i B 〉 N ¯2 B A A 〈 i ∗A i A 〉 N 〈 i ∗B i B 〉 N (3.4a) , (3.4b) where i ′A,B = i A,B − i 0 is the uncorrelated noise component of each diode. In the ¯ ¯ limit that | i 0 | ≪ ¯ i ′ ¯ (that is, we are trying to resolve a small coherent component A,B buried underneath shot noise), then the middle two terms will always be subdominant to the last term, regardless of N , so we can ignore them. Upon each noise realization, the vectors i 0 , i ′A , and i ′B have magnitudes which √ √ √ are drawn from Rayleigh distributions with means S 00 , S A ′ A ′ , and S B′ B′ , respectively. Their phases are all random. However, the two products i ∗0 i 0 and i ′∗ i ′ behave differently upon summation over a large number N of noise realA B izations. Since i ∗0 i 0 is always real (i.e., the phase is always 0), the summation ∑ (k) ∗ (k) i ′B has a random phase, the i 0 grows like N . On the other hand, since i ′∗ k i0 A p ∑ ∗ summation k i ′A(k) i ′B(k) grows like N . Hence, after a large number of averages, √ 〈 i ∗0 i 0 〉 N → S 00 and 〈 i ′∗ i ′ 〉 → S A ′ A ′ S B′ B′ /N . Therefore, these two terms will be A B N 2 equal when N = S A A ( f )S BB ( f )/S 00 ( f ) = 1/γ2AB . So at least 1/γ2AB averages are re- quired to resolve a coherence γ2AB in this case. 3.2 Thermal noise The fluctuation–dissipation theorem The fluctuation–dissipation theorem is a general result in statistical mechanics that quantifies how losses in a system give rise to fluctuations of the system’s coordinates. A derivation of this theorem was first given by Callen and Welton.75 The theorem has found wide use in various areas of precision measurement. In gravitational wave physics, it provides important results for thermally driven noise in mirrors, suspensions, and electronics. Consider a system with some generalized coordinate x, and a corresponding generalized force F . If an oscillatory force F(t) = F0 ( f ) cos(2π f t) is applied to the system at frequency f , the system’s coordinate will respond with a velocity ẋ(t) = 58 ẋ0 ( f ) sin[2π f t + ϕ( f )], where the amplitude ẋ0 ( f ) and phase lag ϕ( f ) are determined by the system’s equation of motion. Often, ϕ( f ) arises from some kind of loss mechanism; for example, mechanical, electrical, or thermodynamic dissipation in the system. The fluctuation–dissipation theorem says that the time-averaged power dissipation ⟨ ⟩ W( f ) = F(t) ẋ(t) causes stochastic fluctuation S xx ( f ) in the coordinate x:a S xx ( f ) = 2k B T W( f ) . π2 f 2 F0 ( f )2 (3.6) Note that ⟨ ⟩ ⟨ ⟩ W( f ) = F(t) ẋ(t) = F0 ẋ0 cos(2π f t) sin(2π f t + ϕ) = 21 F0 ẋ0 sin ϕ ≃ 12 F0 ẋ0 ϕ, (3.7) where the approximation holds for ϕ ≪ 1. This shows that the energy dissipated (and hence the amount of fluctuation) depends on the phase lag ϕ. Equivalently, if we can express the system’s equation of motion in the Fourier domain, then we can extract the transfer function Y ( f ) = ẋ( f )/F( f ) (commonly called the admittance) which relates the complex amplitudes ẋ( f ) = ẋ0 ( f )eiϕ( f ) and F( f ) = F0 ( f )eiψF ( f ) . Then by computing W( f ), we find ¯ 2k B T ¯ ¯ k B T ¯¯ ¯Im χ( f )¯, ¯= Re Y ( f ) πf π2 f 2 / / where χ( f ) = x( f ) F( f ) = Y ( f ) (2πi f ) is the system’s susceptibility. S xx ( f ) = (3.8) Types of thermal noise The fundamentals of mechanical and thermodynamical thermal noises are described by Saulson.77 A more detailed overview of thermal noises relevant to gravitational wave detection is given by Narwodt et al.78 Brownian noise Brownian noise refers to fluctuation arising from friction in a mechanical system. Such friction may arise from rubbing, gas damping, or—of particular concern to lownoise mechanical experiments—internal friction within the bulk or on the surface a This is the classical version of the fluctuation–dissipation theorem, valid in the limit k T ≫ B 2π× f . The fully quantum version of the theorem is obtained by the substitution76 kB T → 2π× f . 1 − exp (−2π× f /k B T) (3.5) 59 of the system. In solid materials like glasses, metals, or plastics, internal friction is quantified by the material’s mechanical Q factor or, equivalently, the loss angle ϕ = 1/Q (which may be frequency-dependent). This loss angle produces a phase lag in the system’s response to an applied force, and therefore produces a loss in accordance with equation 3.6. In the particular case that ϕ is frequency-independent, the loss is called structural. If ϕ ∝ f , the loss is called viscous. Thermodynamic noise Every macroscopic system in thermodynamic equilibrium continually wanders between the many different microstates available to it. This wandering leads to random fluctuations in the system’s state variables, such as temperature, entropy, pressure, volume, and so on. In particular, temperature fluctuations will propagate to fluctuations in other material parameters—such as volume, refractive index, elasticity, and so on—because of the temperature dependence of these parameters.79 The effects of these fluctuations on the optical readout of a mechanical system due to temperature fluctuations are collectively referred to as thermodynamic noise. In particular, • fluctuation from the coefficient of thermal expansion α = (1/L) ∂L/ ∂T is called thermoelastic noise; • fluctuation from the coefficient of thermorefraction β = ∂n/ ∂T is called thermorefractive noise; • fluctuation from α and β considered together coherently is called thermo-optic noise; and • fluctuation from the fractional temperature dependence (1/E) ∂E/ ∂T of the Young modulus E is also said to contribute to thermoelastic noise. Johnson–Nyquist noise Johnson–Nyquist noise80,81 refers to the thermal noise in an electronic circuit arising from its lossy (that is, resistive) components. Consider the simple case of computing the charge fluctuation (or equivalently, the voltage fluctuation) across a capacitor with capacitance C whose plates are connected by a resistor with resistance R . From elementary electrodynamics, we know that if a voltage V is applied to the capacitor, then the charge q on the plates 60 amounts to a stored energy U = qV . Therefore, we can view q as the coordinate for the system and V as the corresponding force, and we should be able to use the fluctuation–dissipation theorem to calculate the stochastic fluctuations in q (or V ) even in the absence of an applied voltage. The system’s admittance can be read off from the system’s equation of motion, which in this case is already known from elementary circuit analysis: q̇( f ) = Y ( f ) V ( f ) = 1 1 + 2πi f RC V(f ) = V ( f ). ZR ∥ ZC R (3.9) We now suppose we are only interested in the effect of the resistor; that is, we stick to frequencies f ≪ 1/RC , where the dynamics of the capacitor does not affect the behavior of the circuit. Then Y ( f ) ≃ 1/R . Then from the fluctuation–dissipation theorem, the charge fluctuation across the capacitor in the absence of an applied voltage is S qq ( f ) = ¯ k B T/R k B T ¯¯ Re Y ( f )¯ = 2 2 , 2 2 π f π f (3.10) and the voltage fluctuation is SV V ( f ) = S qq ( f ) |Y ( f )/2πi f |2 = 4k B TR, (3.11) which is the usual expression for Johnson–Nyquist noise in terms of the voltage fluctuation across the resistor. In suspensions Suspension thermal noise has both Brownian and thermoelastic contributions, which have been characterized by Cumming et al.82 in a modal analysis that considers the suspension fibers, the test mass ears, and the welds connecting the fibers to the ears. Brownian noise Each element of the suspension has Brownian dissipation in the bulk of the glass, and on the glass surface. The bulk loss is very small and nearly viscous, with a loss ( ) angle ϕb = 1.2 × 10−11 × ( f /1 Hz)0.77 . The surface loss is structural, and depends on the intrinsic loss angle ϕs of the surface material, the depth h of the surface, and the thickness d of the suspension element: the loss is 8hϕs /d ≃ 1.2 × 10−7 for d = 400 μm, which is the thinnest element (highest surface-to-volume ratio) of the suspension.82 61 Thermodynamic noise The thermoelastic loss angle ϕ(susTE) of each suspension element depends on contributions from the CTE α of the silica, and the temperature dependence ∂E/ ∂T of its Young modulus. At each point z along the fiber, the loss is given by82 ( ) σ(z) ∂E ω τ(z) ET (susTE) ϕ (ω, z) = α− 2 × , C E ∂T 1 + [ω τ(z)]2 (3.12) where σ(z) is the static stress in the fiber, C is the volumetric heat capacity of the silica, and τ(z) is the thermal time constant of the suspension element, which depends on its material properties and geometry. The negative sign between the thermoelastic and Young modulus terms in equation 3.12 indicates that the thermoelastic loss can be minimized by engineering the appropriate amount of static stress σ. In the Advanced LIGO suspensions, thermoelastic cancellation in the fiber is achieved by making the end portions of the fiber twice the diameter of the central section (800 μm versus 400 μm).82 The main suspension thermal noise contributions to differential arm length fluctuation are from the longitudinal and bounce modes, since these modes produce motion in the beamline direction. In substrates Brownian noise Levin83 computed the Brownian noise for a cylindrical mirror substrate of radius R interrogated by a Gaussian beam with spot size w ≪ R ; this was later extended by Bondu et al.84 and Liu and Thorne85 for cases where w ≪ R is violated. The Levin expression is 4k B T S (subBr) (f ) = xx f p 2 1 − σ2 ϕ I, π3 Ew (3.13) where E is the Young modulus, σ is the Poisson ratio, ϕ is the loss angle in the bulk, and I ≃ 1.9 is a numerical factor. For Advanced LIGO, with w ≃ 6 cm and R = 17 cm, the corrections of Bondu et al. and Liu and Thorne amount to a substrate Brownian noise that is roughly 70 % of the Levin expression in terms of amplitude. The bulk ( ) ( / )0.77 86 loss angle is assumed to be ϕ = 7.6 × 10−12 × f 1 Hz . Additionally, Brownian loss at the substrate surface can be treated in the same way as coating loss, with a thickness–loss product of d ϕs = 5 × 10−12 m.86 62 Thermodynamic noise The thermoelastic noise of test mass substrates have been described by Braginsky et al.79 and Cerdonio et al.87 This noise (per test mass) is given by S (subTE) (f ) = xx where88 4k B T 2 α2 (1 + σ)2 w ( / ) J f fT , κ π1/2 { iΩ/2 [ 1/2 ]} e (1 − iΩ) Ω (1 + i) 1 1 J(Ω) = − Re erfc + − 2 Ω2 Ω2 (πΩ3 )1/2 (3.14) (3.15) and f T = κ/πw2 C , and the material parameters refer to the substrate. The function p J(Ω) has asymptotes 1/ 8Ω for Ω ≪ 1 and 1/Ω2 for Ω ≫ 1. In coatings Brownian noise Brownian noise in coatings is analyzed in detail by Hong et al.,89 who derive a general expression for the noise in a multilayer dielectric stack interrogated by a laser beam. In general, any isotropic and homogeneous material has two elastic loss angles: a loss angle ϕK corresponding to the bulk modulus K , and a loss angle ϕG corresponding to the shear modulus G . However, these individual loss angles are (so far) poorly constrained in the coating materials used in GW detection, and are usually assumed to be equal. The Advanced LIGO coatings consist of multilayer stacks of silica (SiO2 ) and titaniadoped tantala (Ti:Ta2 O5 ). Silica has a loss angle of about 5 × 10−5 , while titaniadoped tantala has a much larger loss angle—about 3 × 10−4 .90 The tantala loss is the dominant thermal noise contributor to the Advanced LIGO strain sensitivity. Measurements by Gras et al.90 on Advanced LIGO coating witness samples indicate that the expected coating thermal noise contribution to the differential arm length sensitivity is 1/2 = S (cBr) xx ( f ) ( −20 1.15 × 10 1/2 m/Hz ) ( ) 100 Hz 1/2 × . f (3.16) Thermodynamic noise Thermodynamic noise in coatings is computed by finding the spectrum S TT of temperature fluctuations corresponding to a Gaussian pressure profile.79,85,91–93 This 63 spectrum is 23/2 k B T 2 S TT ( f ) = πκs w [ ∫∞ du Re ( 0 with f T = κc /πC c w2 . The integral goes to ] 2 ue−u /2 / )1/2 , u2 − i f f T (3.17) √ / p π/2 for f ≪ f T , and to f T 2 f for f ≫ f T . For the Advanced LIGO coatings, κc ∼ 10 W/(m K) and C c ∼ 1 J/(m3 K), so for w ≃ 6 cm we have f T < 10−4 Hz. Evans et al.92 then considered thermo-optic noise by propagating S TT ( f ) to reflected phase fluctuations arising from the coefficient of thermal expansion α = (1/L) ∂L/ ∂T and coefficient of thermorefraction β = ∂n/ ∂T for the coating layers. The resulting reflected phase noise (referred to effective mirror displacement) is )2 ( S xx ( f ) = S TT ( f ) Γ( f ) αc d − βc λ − αs dC c /C s , (3.18) where • αc , βc , and C c are the CTE, CTR, and heat capacity of the coating; • αs and C s are the CTE and heat capacity of the substrate; • d is the coating thickness; and • Γ is a frequency-dependent factor accounting for the nonzero thickness of the coating. Importantly, the relative sign of the αc and βc is negative, indicating that an uncorrelated analysis of the thermoelastic and thermorefractive noises overestimates the thermo-optic noise for positive αc and βc (as is the case for silica/tantala coatings). In fact, the coating layer structure can be optimized to cancel the thermoelastic and thermorefractive terms, thereby suppressing the total thermo-optic noise.94 The Advanced LIGO coatings were not optimized in this way, but nonetheless some thermo-optic cancellation is present. For Advanced LIGO, the expected thermo-optic noise contribution to the differential arm length sensitivity does not exactly follow a power law. It is expected to be 4.1 × 10−21 m/Hz1/2 at 10 Hz and 2.2 × 10−21 m/Hz1/2 at 100 Hz.92 Photothermal noise For a homogeneous test mass, the photothermal transfer function, taking incident power to mirror surface displacement, is79,87,95 x( f ) α(1 + σ) A =− 2 2 × , P( f ) π w C if (3.19) 64 glass optic plastic screw metal holder Figure 3.2: Cross-sectional diagram of the signal recycling mirror holder. The SRM is 5.08 cm in diameter and is held inside an aluminum ring with two PEEK set screws. valid for f ≫ f T ≡ κ/πCw2 ≃ 0.1 mHz. The absorptivity of the test mass coatings is excellent, with typical values that are a few hundred parts per billion. With α = 5.1 × 10−7 K−1 , σ = 0.17 C = 1.6 × 106 J/(m3 K), and w = 5 cm, this implies a transfer function magnitude ) ( ) ( ¯x¯ ( ) A 100 Hz ¯ ¯ −19 × . (3.20) m/W × ¯ ¯ = 5 × 10 P f 1 ppm p With Parm = 100 kW, this implies a shot noise 2hνP ≃ 2 × 10−7 W/Hz1/2 , this implies a photothermal shot noise per test mass of 2.5 × 10−26 m/Hz1/2 for A = 250 ppb. This is completely negligible for Advanced LIGO even at design sensitivity. Even with excess intensity noise in the arms arising from a RIN of 10−8 /Hz at the interferometer input (figure 3.5), the resulting photothermal noise is 7.5 × 10−25 m/Hz1/2 per test mass at 100 Hz, which is still negligible. A more exact expression for the test mass photothermal transfer function, including terms accounting specifically for coating thermo-optic noise and substrate thermoelastic noise, is given by Farsi et al.;95 however, this will not bring the expected photothermal noise (either from shot noise or classical intensity noise) to a level which limits Advanced LIGO. Elsewhere During O1, the signal recycling mirror was unique among the interferometer optics in that it was set inside a metal holder using two 8–32 PEEK plastic screws 65 (figure 3.2). Finite element analysis of the assembly indicated the shear vibration of these screws should have a resonance at about 3 kHz,96 and indeed a resonance in the transfer function from SRM motion into POP45 light was seen at 3.3 kHz in H1 and 2.4 kHz in L1.97,98 This frequency is consistent with an effective free vibration length ℓ of roughly 0.9 mm: the shear compliance per screw is x/F = ℓ/G A , ( )2 where G ≃ 1.4 GPa is the shear modulus for PEEK and A = π 4.2 mm = 54.5 mm2 is the area of each screw. With an effective mass m = 100 g for this mode, the resonant p / frequency is then f 0 = G A/2 ℓm 2π. The width of the resonance in H1 implies a Q factor of 170, or equivalently a loss angle ϕ = 1/Q = 6 × 10−3 , assuming structural damping. This implies a Brownian SRM motion of77 /( 2 ) 2 2k B T ϕ 4π m f 0 S xx ( f ) = , (3.21) ( ) π f 1 − f 2 / f 2 2 + ϕ2 0 ( ) ( / )1/2 which amounts to an SRM displacement of 6 × 10−17 m/Hz1/2 × 100 Hz f for f ≪ f 0 . The agreement between the expected peak height from equation 3.21 and the observed peak height in the displacement-calibrated SRC length error signal is excellent,99 as is the agreement between the expected peak height and the observed height in the differential arm length readout (using the measured coupling shown in figure 2.18). Note, however, that the behavior of the noise around the resonance frequency cannot constrain the slope of the loss angle, so the assumption here that the loss is structural has not been confirmed. The thermoelastic contribution from the plastic is expected to be small: most of the energy of the mode is stored in shear strain, while thermoelastic noise arises only from bulk strain. 3.3 Seismic and Newtonian noise Seismic noise Ground motion is subject to daily and seasonal variations, particularly for frequencies below 1 Hz.100 However, for quiet above-ground sites like Hanford and Livingston, the motion above 1 Hz in each direction has the approximate displacement ASD101 ) 1 Hz 2 × S( f ) = 10 m/Hz . (3.22) f Coupling of this noise into differential arm length comes from ground motion in ( −8 1/2 ) ( the beamline direction for each test mass, and from ground motion in the vertical direction that couples to beamline motion because the earth’s curvature (see 66 section 2.6). Below the GW band, seismic noise is suppressed by an active seismic isolation system,40 and within the GW band the noise is passively filtered by the mechanical transfer functions of the mirror suspensions. Newtonian noise Newtonian noise refers to displacement caused by fluctuations in the gradient of the Newtonian potential Φ(r) at each test mass. Such gradient fluctuations can be sourced by seismic noise, vibration of structures and equipment nearby the masses, and (of lesser concern to current gravitational wave detectors) fluctuations in atmospheric pressure.102 In future observing runs, the effect of Newtonian noise on the differential arm length readout may be estimated using seismometer arrays. This effect can then be subtracted from the differential arm length signal, either online or offline.103 3.4 Gas noise The LIGO vacuum system is designed to be held at about 0.1 μPa, with the dominant species of residual gas being H2 .104 The small amount of residual gas leads to two kinds of gas noise; these are included in the budget in figure 3.12. Optical scattering The first kind of gas noise is a sensing noise induced by gas molecules wandering across the beam path and scattering light out of the beam. This noise has a flat phase spectrum up to a cutoff frequency that depends on the species of the gas.101,105 For some particular species of gas, the displacement-referred noise induced along a single arm is (4πα)2 n S xx ( f ) = v ∫L exp[−2π f w(z)/v] , dz w(z) (3.23) 0 where α, n, and v are the polarizability, number density, and speed of the gas, and w(z) is the spot size along the arm. 67 Squeeze film damping The second kind of gas noise is a displacement noise induced by gas molecules hitting the test masses.106,107 This is particularly problematic for the Advanced LIGO end test masses because of the small gap (5 mm) between the test mass and the reaction mass. A single molecule may bounce between the masses many times as it passes through the gap; this is referred to as “proximity-enhanced gas damping” or “squeeze-film damping”. For a given test mass geometry, the presence of the gas induces a mechanical impedance Z( f ) = F( f )/ ẋ( f ) that is proportional to the gas pressure p and the time τ required for the gas to diffuse into or out of the gap. Additionally, the impedance has a pole at 1/τ. The full expression for the force noise from squeezed-film damping is well-modeled by the formula106 ∞) S (sfd) ( f ) = S (FF + FF ∆S FF ( )2 , 1 + 2π f τ (3.24) ∞) where S (FF is a white force noise that accounts for molecular collisions against the test mass in the absence of the reaction mass, and ∆S FF accounts for the proximityenhancement effect. To accurately compute the quantities in equation 3.24, numerical simulations are required (and the results are quoted below). However, analytical calculations by Dolesi et al. yield approximate expressions for τ and ∆S FF :106 ( ) π m g 1/2 R 2 /d [ ] τ= 2k B T ln 1 + (R/d)2 pτ/d ∆S FF = 4k B T × πR 2 × ( )2 , 1 + 2π f τ (3.25) (3.26) where T is the temperature, R is the test mass radius, d is the gap size, p is the pressure, and m g is the mass of the gas species. Numerical simulations of the Advanced LIGO end test mass geometry indicate that for 1 μPa of H2 , the expected force spectrum for a single end test mass is flat up to √ 1/2πτ = 1/(2√ π × 800 μs) = 200 Hz, with a dc value ∆S FF = 1.2 × 10−14 N/Hz1/2 . The ∞) ac value is S (FF = 1.4 × 10−15 N/Hz1/2 .108 We can use the known compliance of the test masses to convert the force spectrum into a displacement spectrum. The displacement noise from squeeze-film damping for two independent end test masses in the presence of residual H2 is then ( ) ( ) ( ) p H2 30 Hz 2 −20 1/2 (sfd) m/Hz × for f < 200 Hz. (3.27) S xx ( f ) = 1.2 × 10 × 1 μPa f 68 3.5 Frequency noise Budget of frequency noise in the interferometer In this section we discuss the noises contributing to the frequency stabilization system. The topology of the frequency stabilization loop was discussed in detail in section 2.2, and a block diagram reduction of the loop was given in figure 2.9. Our objective in this section is to produce a budget of the residual frequency noise r at the input of the interferometer; this input frequency noise is filtered by the common-mode cavity and then propagates to the differential arm length readout via a coupling function presented later. A loop analysis will show that [( ) ] − A F + K M P n p + F n f + K Mn m + K n k + n a − n g 1 [( ) ] r= × , 1 + i f / f IMC A F +KM P +K +1 (3.28) where n X refers to noise injected immediately before block X in the figure 2.9. The budget of noises considered in this section is shown in figure 3.3. Noise in the laser The typical freerunning noise of the NPRO (including the injection-locked oscillator) ( ) ( / ) is 100 Hz/Hz1/2 × 100 Hz f .36 At sufficiently high frequencies, this noise eventually becomes subdominant to the Schawlow–Townes limit109 √ hν0 1 (ST) S νν (f ) = , 2πτ P (3.29) which is a fundamental limit depending only on the laser frequency ν0 , the output power P , and the storage time τ of the laser resonator. For a resonator roundtrip length L ∼ 10 cm and a coupler transmissivity T ∼ 1 %, the storage time is τ = −2L/c ln(1 − T) ∼ 70 ns. For P ∼ 1 W of output, the Schawlow–Townes limit is (ST) then S νν ( f )1/2 ∼ 1 mHz/Hz1/2 , which is subdominant to the aforementioned 1/ f noise below 1 MHz. Noise in reference cavity The goal of the reference cavity loop is to stabilize the laser frequency noise down to the (apparent) length noise of the fused-silica reference cavity. The performance 69 carm sensing and displacement Frequency actuation imc sensing imc displacement fss noise [rough] Total £ ¤ asd of frequency Hz/Hz1/2 10−4 10−5 10−6 10−7 10−8 10−9 101 102 103 104 Frequency [Hz] Figure 3.3: Budget of frequency noise of the light at the input of the Advanced LIGO Hanford detector. of the Advanced LIGO reference cavity stabilization system was investigated extensively by Chalermsongsak110 and Chalermsongsak et al.32 in a test setup at Caltech. At best, this stabilization system should be limited by the Brownian noise ( ) ( / )1/2 of the reference cavity mirrors, about 6 mHz/Hz1/2 × 100 Hz f . However, even in the test setup this performance was degraded by a number of technical noise sources below a few tens of hertz, with the largest technical noise contributors being scattered light and seismic motion. Additionally, the apparent length noise at a few hundred hertz is additionally susceptible to acoustic noise.32 Therefore, one might say more conservatively that in the presence of some common technical issues, the reference cavity stabilization loop is able to stabilize the laser frequency to better than 0.1 Hz/Hz1/2 above 30 Hz. One must also consider the frequency noise on the VCO used to drive the reference cavity AOM. Modeling111 and measurement112 indicate that the double-pass frequency noise is better than 6 mHz/Hz1/2 above 10 Hz. 70 Noise in the input modecleaner In the GW band, the frequency stabilization to the input modecleaner should be limited by the shot noise of the light on the PDH photodiode. With 25 W of input light into the modecleaner, about PIMC = 50 mW is incident on the PDH photodiode with the modecleaner unlocked. Assuming the reflected light from the locked modecleaner is dominated by sideband light, and assuming ΓIMC = 0.01 rad, this sets the √ shot noise limit at ( f IMC /2) 3hc/λ0 PIMC = 15 μHz/Hz1/2 . Because the bandwidth of the IMC loop is tens of kilohertz, the loop impresses this shot noise onto the laser frequency. In practice, because of imperfect modematching into the modecleaner, a few milliwatts of carrier light remains on the PDH detector even when the modecleaner is locked, making the shot-noise-limited frequency spectrum more like a few parts in 10−4 Hz/Hz1/2 . Noise in the common-mode and PRC lengths The noise performance of this loop is limited by the shot noise on the PDH photodiode, along with the photodiode dark noise and the electronics noise of the first few electronic amplifiers. With 25 W of power at the interferometer input, these sources together limit the frequency noise performance to roughly 1 μHz/Hz1/2 between 10 and 100 Hz, and roughly 10 μHz/Hz1/2 at 1 kHz. The interferometer passively filters this input frequency noise above the common-mode pole f + ≃ 0.6 Hz, attenuating the frequency noise on the circulating light to less than 10−8 Hz/Hz1/2 at 100 Hz. The 9.1 MHz reflection PDH sensor additionally senses PRC length fluctuation, as seen in equation 2.21; therefore, PRC length fluctuation acts as an error point offset for the common-mode arm length stabilization loop. Coupling into differential arm length readout The basic coupling mechanism can be understood by considering an asymmetric Michelson with differential mirror reflectivities: ] E in (ω) [ E out (ω) = r X e2iω ℓX /c + r Y e2iω ℓY /c 2 [ ] = E in (ω)e2iω ℓ+ r + cos(2ω ℓ− /c) + ir − sin(2ω ℓ− /c) , (3.30) with r ± = (r X ± r Y )/2 and ℓ± = ( ℓX ± ℓY )/2. The cosine term in brackets in equation 3.30 is the usual Michelson response given a certain amount of length offset ℓ− . This 71 Magnitude [rin/Hz] 10−1 10−2 10−3 10−4 10−5 10−6 101 102 103 101 102 103 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 3.4: Frequency noise coupling into the differential arm length readout. offset may include a macroscopic Schnupp offset ℓ− , as well as a microscopic dc readout offset δ ℓ(0) − . The sine term in brackets in equation 3.30 is the “contrast defect” light. It converts phase sidebands at the interferometer input into amplitude sidebands at the output: a sideband of the form ±iΓei(ω0 ±ω)t at the input emerges as ∓Γ r − sin[2(ω0 ± ω) ℓ− /c]ei(ω0 ±ω)t at the output. Importantly, in the plane-wave approximation, and in the absence of radiation pressure, both a reflectivity imbalance and a length imbalance are required to produce a contrast defect. Advanced LIGO has two (intentional) sources of length imbalance; namely, the macroscopic Schnupp asymmetry in the Michelson length and the microscopic dc readout offset in the differential arm length. There are also (unintentional) arm reflectivity imbalances, which can manifest as dc reflectivity imbalances or as imbalances in the arm pole frequencies. If radiation pressure is included, then a length imbalance alone is sufficient to pro- £ ¤ ASD of relative intensity noise 1/Hz1/2 72 In-loop Out-of-loop 10−5 10−6 10−7 10−8 10−9 10−10 100 101 102 103 Frequency [Hz] Figure 3.5: Relative intensity noise at the interferometer input, as measured by the in-loop and out-of-loop photodiodes for the outer intensity loop. duce a frequency noise coupling into differential arm length. In Advanced LIGO, the dc readout offset in the differential arm length causes laser frequency fluctuations to convert to differential arm power fluctuations, since each arm is slightly detuned from resonance.21 Approximate analytical expressions for frequency coupling into differential arm length readout have been computed by Izumi and Sigg,20,21 taking into account the aforementioned length and reflectivity imbalances, radiation pressure, and also a static common-mode arm length offset. The measured H1 frequency noise coupling into the differential arm length readout is shown in figure 3.4. 3.6 Intensity noise The out-of-loop sensor of the outer loop measures the amount of RIN at the interferometer input. A plot of this RIN is given in figure 3.5. Coupling into differential arm length readout Approximate analytical expressions for intensity coupling into differential arm length readout have been computed by Izumi and Sigg.18,20,21 The intensity cou- 73 Magnitude 100 10−1 10−2 10−3 101 102 103 104 102 103 104 180◦ 135◦ Phase 90◦ 45◦ 0◦ −45◦ −90◦ −135◦ −180◦ 101 Frequency [Hz] Figure 3.6: Intensity noise coupling into the differential arm length readout, in terms of input RIN into antisymmetric-port RIN. pling arises through a few different mechanisms. 1. dc readout: Input intensity fluctuation modulates the amount of dc readout light exiting the interferometer. 2. Contrast defect: Similarly, input intensity fluctuation modulates the amount of contrast defect light exiting the interferometer. As with the frequency noise coupling, the magnitude of this effect depends on arm imbalances such as Schnupp asymmetry and differential frequency-dependent reflectivity. 3. Radiation pressure: Input intensity fluctuations produce radiation pressure fluctuations in the arms. If the arms are perfectly balanced, this fluctuation is purely common-mode. However, any of the aforementioned arm imbalances (as well as an imbalance in the reduced masses of the arms) will produce a differential signal. 4. rf sidebands: Because the attenuation of the rf sidebands through the OMC is finite (T = 60 ppm for the 45.5 MHz sidebands), intensity noise of the sideband 74 aLIGO design Dielectric noise Resistor damping Total £ ¤ ASD of displacement m/Hz1/2 10−17 10−18 10−19 10−20 10−21 101 102 103 Frequency [Hz] Figure 3.7: Noises associated with the test mass electrostatic actuator, along with the Advanced LIGO design sensitivity. The noises shown here scale with the bias voltage Vb applied to the test mass. light can create signal on the GW photodiodes. Furthermore, because the rf sidebands are not resonant in the arms, they are not filtered by the commonmode cavity pole, so input intensity fluctuations light appear directly on the sideband light at the antisymmetric port. The measured H1 intensity noise coupling into the differential arm length readout is shown in figure 3.6. The projection onto the differential arm length readout is then given in figure 3.12. 3.7 Photodiode noise The photodiodes, their readout electronics, and the analog-to-digital conversion produce noise even in the absence of light. This noise amounts to 1.0 × 10−8 mA/Hz1/2 (equivalent to 0.3 mA of photocurrent) per diode, with a 1/ f knee around 60 Hz.113 3.8 Actuator noise In this section we discuss some actuator noises associated with the test masses. Estimates of these noises are shown in figure 3.7. 75 DAC noise The 18-bit DAC used to drive each suspension actuator has a typical voltage noise of114 ( ) (DAC) SV ( f )1/2 = 300 nV/Hz1/2 × V √ ( ) 50 Hz 2 1+ . f (3.31) Only the test mass DACs are expected to have the capacity to significantly impact the differential arm length sensitivity. During O1, the low-noise test mass actuator electronics had a dc gain of 2 V/V, with poles at 2.2 Hz, 2.2 Hz, and 152 Hz, and zeros at 50 Hz and 50 Hz. Combined with a force coefficient of 2αVb ≃ 2 × (2 × 10−10 N/V2 ) × (380 V) = 1.1 × 10−7 N/V and a mechanical compliance of 2.6 × 10−3 m/N × (0.4 Hz/ f )2 , this implies a DAC-induced displacement noise of 1.2 × 10−22 m/Hz1/2 at 50 Hz for each test mass, rising like f 3 below 50 Hz. Electrostatic damping Dielectric loss In this section we examine noise arising from dielectric loss in the test mass substrate, which couples into test mass displacement via the electric field of the electrostatic driver. The principle of the electrostatic driver is that an electric field E applied to the test mass creates a polarization density P = ε0 χE in the substrate. Here χ is the susceptibility of the substrate. To include the dielectric loss, we add a small imaginary component to the susceptibility: χ = χ0 (1 + iϕχ ). Equivalently, if the test mass and the electrostatic driver are viewed as a capacitor, then the dielectric loss of the substrate makes the capacitance complex: C = C 0 (1 + iϕC ). The dielectric loss of Suprasil glasses is quoted as 5 × 10−4 at 1 kHz,b although other measurements on fused silica indicate a much lower audio-band dielectric loss—less than 4 × 10−6 .c If a significant fraction of the capacitance comes from the field in the substrate (rather than field in the vacuum gap between the masses), then the loss angle of the capacitance will nearly equal the loss angle of the substrate. b Heraeus, Quartz Glass for Optics: Data and Properties, HQS-MO_01.4/E/07.2015 c Dynasil quotes this measurement as being “performed at Laboratory of A. R. Von Hippel, Laboratory for Insulation Research, Massachusetts Institute of Technology, May 1970.” (http: //www.dynasilfusedsilica.com/techinfo.phtml?tid=9) 76 If a sinusoidal voltage V (t) = V0 cos(ω t) is applied to the ESD, a charge q will ac[ ] cumulate on the electrodes, with q(t) = C 0 V0 cos(ω t) + ϕC sin(ω t) . Then the timeaveraged power dissipation in the dielectric is ⟨ ⟩ W = q̇V = 21 C 0 V02 ϕC ω, (3.32) and from the fluctuation–dissipation theorem, the PSD of voltage fluctuation is therefore ¯ ¯ ¯ V ( f ) ¯2 2k B T W( f ) 2k B T ϕC ¯ ¯ × SV V ( f ) = ¯ = . q( f ) ¯ π f C0 π2 f 2 V02 (3.33) Given a drive voltage V on the electrode, the force exerted on the test mass is [ ] 1 ∂C 2 ∂U ∂ 1 2 F =− =− C(x)V = − V . (3.34) 2 ∂x ∂x 2 ∂x The coefficient α = 12 ∂C/ ∂x is known from measurement to be about 2 × 10−10 N/V2 . Given some static bias voltage Vb , the small-signal voltage-to-force coefficient is then ( ∂C/ ∂x)Vb ≃ 8 × 10−8 N/V for Vb = 380 V. This measurement can also be used /( ) to infer the capacitance of the system: assuming C 0 (x) = 2α d 1 + x/d , we have ¯ ∂C/ ∂x¯ ≃ 2α, so C 0 (0) ≃ 2 pF for d = 5 mm.d These numbers are sufficient to com0 pute S V V ( f ) and S FF ( f ). S xx ( f ) is then computed from S FF ( f ) via the free-mass compliance 1/M ω2 , with M = 40 kg. The resulting estimate for the displacement noise from dielectric loss is 1/2 S xx ( f ) ( = 9.2 × 10 −24 m/Hz 1/2 ) ( × ϕC )1/2 1 × 10−6 ( ) ( ) Vb 100 Hz 5/2 × × , 100 V f (3.36) ( ) ( / )5/2 which gives S xx ( f )1/2 = 7.0 × 10−23 m/Hz1/2 × 100 Hz f for ϕC = 4 × 10−6 and Vb = 380 V. This noise term is shown in figure 3.7. This estimate does not consider effects from nearby conductors (for example, the ring heaters) or other capacitive effects from the cables connecting the electrodes to the flange. d Numerical simulations by Evans and Miller115 instead suggest C 0 (x) ≃ 2α d/b (1 + x/d)b with b = 1.38, but quantitatively this does not affect the result too much. (3.35) 77 Circuit damping We now examine noise arising from the circuit that drives the reaction mass electrodes.116,117 Between each amplifier and each electrode there is a series resistor that protects against overcurrent. Each resistor injects Johnson–Nyquist noise (that is, charge fluctuation) into the electrodes, which adds damping to the test mass. Assuming the bias electrode is held at Vb = V0 and each signal electrode is held at 0 V, the steady-state relationship between the bias voltage and the charge on the electrode is V0 = q 0 /C 0 . Here we assume (again) that the position-dependent / capacitance is given by C(x) = C 0 (1 + x/d), where d = 5 mm is the position of the mass when the bias is 0 V. We’ll compute the Q of the pendulum mode in the presence of damping by first computing the peak energy stored in the mode, and the energy dissipated per cycle in the resistor. The position dependence of the potential energy of the total electromechanical system is U(x) = 21 C(x)V02 + 12 µω20 x2 α dV02 (3.37a) + 1 µω2 x2 1 + x/d 2 0 ( ) x x2 2 ≃ α dV0 1 − + 2 + 12 µω20 x2 d d (3.37b) = 12 k(x − x0 )2 + U0 (3.37d) ≡ 12 kξ2 + U0 , (3.37e) ξ = x − x0 (3.38) = (3.37c) where 2αV02 + µω20 ≃ µω20 (3.39) x0 = αV02 /k ≃ αV02 /µω20 , (3.40) k= d and U0 is a constant energy which we ignore henceforth. Note that the approximation in equations 3.39 and 3.40 is good to about 100 ppm for Advanced LIGO; in other words, the presence of the charged electrode does not significantly alter the pendulum dynamics. 78 Suppose the pendulum oscillates by an amount ξ(t) = Ξ cos ω t. The peak energy stored is U(Ξ) = 12 kΞ2 . To compute the energy ∆U dissipated as the pendulum swings, we need to compute the current flowing through the resistor. First, we compute the charge stored on the capacitor: ( ) x x2 q = C(x)V0 ≃ 2α dV0 1 − + 2 . d d Therefore, the current flowing across the resistor is ( ) 2x q̇ = 2αV0 ξ̇ − 1 ≃ −2αV0 ξ̇, d (3.41) (3.42) where we have used the facts that ẋ = ξ̇ and x/d ≪ 1. The energy dissipated in the resistor over one cycle 2π/ω of the pendulum motion is therefore 2π ⟨ 2 ⟩ × q̇ R ω ⟨ ⟩ 2π = × (2αV0 )2 ξ̇2 R ω 2π (ωΞ)2 = × (2αV0 )2 × ×R ω 2 = πω(2αV0 Ξ)2 R, ∆U(Ξ) = (3.43a) (3.43b) (3.43c) (3.43d) and therefore the Q factor of the motion is Q(ω) = 1 2 µω20 U(Ξ) 2 kΞ = = . ∆U(Ξ) πω(2αV0 ωΞ)2 R 2πω(2αV0 )2 R (3.44) Since 1/Q ∝ ω, the damping is viscous. Setting ω = ω0 gives the Q for the pendulum mode: Q(ω0 ) = µω0 2π(2αV0 )2 R . (3.45) For µ = 16 kg, ω0 /2π = 0.4 Hz, α = 2 × 10−10 N/V2 , V0 = 380 V, and R = 10 kΩ, the expected Q is then 1.1 × 1010 , which is greater than the expected mechanical Q of the ( ) pendulum alone. However, the shallower slope of the electrical damping 1/ f 2 com( ) pared to the mechanical damping 1/ f 5/2 means that the resistive damping noise overtakes the structural mechanical noise above 10 Hz. The same answer could have been reached more simply by computing the Johnson– Nyquist noise of the electrode resistors and propagating the resulting voltage noise into test mass displacement. However, the above calculation shows that this simplified approach is not guaranteed to work if (1) the electromechanical stiffness k (equation 3.39) is significantly different from the mechanical stiffness µω20 , or (2) the thermal motion of the pendulum is a significant fraction of the gap size, so that the approximation x/d ≪ 1 in equation 3.42 does not hold. 79 Freerunning Residual 10−5 10−6 £ ¤ asd of dipslacement m/Hz1/2 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14 10−15 10−16 10−17 10−2 10−1 100 101 102 103 Frequency [Hz] Figure 3.8: Freerunning and residual length noises for the power recycling cavity. 3.9 Noise from vertex degrees of freedom Figures 3.8, 3.9, and 3.10 show the residual and freerunning displacement noises for the power-recycling cavity length, the Michelson length, and the signal-recycling cavity length. The residual displacement noise of each degree of freedom—particularly the Michelson and SRC lengths—can couple into the differential arm length readout. Above a few tens of hertz, the residual displacement noise comes mostly from sensing noise that is injected into the loops. The effect of this reinjected sensing noise is cancelled in the differential arm length readout using feedforward subtraction for the Michelson and SRC lengths, as described in 2.4. Nonetheless, residual noises in these three degrees of freedom can still couple into the differential arm length readout, potentially because of imperfect feedforward cancellation, the presence of significant displacement noises not arising from sensing noise, or bilinear couplings (which cannot be subtracted with an LTI feedforward scheme). To measure the amount of vertex noise in the differential arm length readout, noise is injected into each vertex degree of freedom, with an amplitude sufficient to produce a clear excess in the differential arm length readout above the quiescent level. 80 Freerunning Residual 10−5 10−6 £ ¤ asd of dipslacement m/Hz1/2 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14 10−15 10−16 10−17 10−2 10−1 100 101 102 103 Frequency [Hz] Figure 3.9: Freerunning and residual length noises for the signal recycling cavity. One could use this data to produce a transfer function from the vertex degree of freedom to the differential arm length (as is shown in figures 2.16, 2.17, and 2.18), and then use the transfer function to propagate the quiescent vertex residual noise into the differential arm length. However, to account for bilinear couplings it is better to compute the excess power ratio between the vertex residual and the differential arm length readout, and use this power-based coupling to propagate the quiescent vertex residual noise into the differential arm length. This is the technique that is used for the noise budget presented in this chapter. 3.10 Noise from angular degrees of freedom The impact of residual angular fluctuations on the differential arm length readout is budgeted in a fashion similar to the vertex length degrees of freedom, as described in section 3.9. Only the differential hard loops, the Michelson loops, and the SRC pointing loops are measured and budgeted, because the other loops are low-bandwidth and their impact on the differential arm length noise is expected to be minimal. 81 Freerunning Residual 10−5 10−6 £ ¤ asd of dipslacement m/Hz1/2 10−7 10−8 10−9 10−10 10−11 10−12 10−13 10−14 10−15 10−16 10−17 10−2 10−1 100 101 102 103 Frequency [Hz] Figure 3.10: Freerunning and residual length noises for the Michelson degree of freedom. 3.11 Jitter noise Jitter coupling was measured by exciting the pointing into the input modecleaner, and using the modecleaner wavefront sensor error signals as a witness for the amount of jitter being injected. The modecleaner suppresses, but does not completely remove, the jitter that is sent into the interferometer. The quiescent level of the wavefront sensor error signals represents an upper limit to the amount of jitter entering the interferometer. 3.12 Noise budget In figure 3.11 we show a budget of fundamental noises limiting the differential arm length sensitivity for H1 during the first observing run. In figure 3.12 we show a budget that includes both the fundamental noises and the known technical noises described previously. ASD of displacement [m/Hz 1/2 ] 82 Quantum Seismic Newtonian Suspension Thermal Coating Brownian Coating Thermo-optic Substrate Brownian Excess Gas Total noise 10 -18 10 -19 10 -20 10 -21 10 1 10 2 10 3 Frequency [Hz] Figure 3.11: Noise budget of Advanced LIGO fundamental noises for the first observing run, computed using GWINC.86 83 aLIGO H1 freerunning DARM, 2015–10–24 13:30:00 Z Measured Quantum noise Dark noise Seismic+Newtonian Thermal Actuator noise Ambient electrostatic £ ¤ ASD of displacement m/Hz1/2 10−17 Gas noise LSC ASC Intensity+Frequency Jitter Total expected 10−18 10−19 10−20 10−21 101 102 103 Frequency [Hz] Figure 3.12: Noise budget of Advanced LIGO Hanford at the start of the first observing run. 84 4 Observational results Advanced LIGO’s first observing run resulted in the first detection of gravitational waves,118 with the second detection following soon after.119 Both events resulted from the coalescence of heavy stellar-mass black holes (on the order of tens of solar masses) merging at redshift z ∼ 0.1. In addition to furnishing the first direct detection of gravitational waves, these events also provide the first evidence for binary black hole systems and binary black hole coalescences.120 This chapter recapitulates the main astrophysical results of the first observing run; these results are the culmination of work performed by the entire LIGO–Virgo scientific collaboration. We will not comprehensively review the great variety of searches that are performed on the strain data, or the technical details of these searches. 4.1 GW150914 GW150914118,122 was a gravitational wave event generated by the coalescence of two black holes, each a few tens of solar masses. The waveform was loud enough that it could be seen by eye in the bandpassed strain time series. The signal spent 8 cycles between 35 and 150 Hz. At 150 Hz, the strain reached its peak of about 1.0 × 10−21 , after which the signal rang down. A time–frequency plot of the signal is shown in figure 4.1, and the noise performance of the Hanford detector during this event (and the subsequent events) is shown in figure 4.2. This event was initially found in low-latency by the burst pipeline,123 and was later also extracted from the data using the compact-binary-coalescence pipeline.124 The parameters of the event (masses spins, and so on) were later estimated using a dedicated parameter estimation pipeline.125 GW150914 enabled new tests of general relativity126 and inferences about the rate of binary black hole mergers.127,128 512 256 128 64 32 Hanford, Washington (H1) 0.30 0.35 0.40 Time (s) 0.45 Normalized amplitude Frequency (Hz) 85 8 6 4 2 0 Figure 4.1: Spectrogram of the GW150914 signal in the H1 detector. Reproduced from Abbott et al.118 4.2 LVT151012 LVT151012122 was a 1.7σ eventa that showed up with an SNR 9.7 between the two detectors. If indeed it came from a binary coalescence, it would correspond to a binary black hole system that is somewhat lighter than GW150914 (a chirp mass ≃ 15 M⊙ ) and somewhat further away ( z ≃ 0.2), with a ringdown frequency of 400 to 500 Hz. 4.3 GW151226 GW151226119,122 was produced by another lighter binary black hole coalescence. Compared to GW150914, the SNR is slightly lower and accumulated over a larger number of cycles: the waveform spent about 45 cycles between 35 and 100 Hz, and then about 10 cycles from 100 to 450 Hz. a The low statistical significance of this signal (< 5σ) is the reason that is designated “LVT” (LIGO– VIRGO trigger) and is not given the “GW” designation. 86 £ ¤ ASD of strain 1/Hz1/2 10 2015–09–14 2015–10–12 2015–12–26 −20 10−21 10−22 10−23 101 102 103 Frequency [Hz] Figure 4.2: Advanced LIGO H1 strain spectra during the three events GW150914, LVT151012, and GW151226. The peaks around 300 Hz are jitter from a piezoelectrically actuated mirror mount that is used to inject the laser light into the vacuum system. These peaks were reduced midway through the observing run by applying epoxy to the mount.121 The change in the noise floor between 40 Hz and 200 Hz occurs in the region where the spectrum is dominated by unknown noise. In addition to the mains line (60 Hz) and harmonics, one can see calibration lines at 35, 332, and 1080 Hz, the test mass roll mode at 13 Hz, a triple suspension roll mode at 41 Hz, test mass violin modes at 500 Hz and harmonics, the OMC length dither line at 4.1 kHz, and the OMC angle dither lines around 2.1 kHz. Noise analysis of these spectra can be found in chapter 3 and the references therein, particularly Martynov et al.43 87 5 Some topics in calibration This chapter aims to introduce some concepts and computations that will be useful for current and future calibration efforts. • In section 5.1 we give a brief overview of why and how GW interferometers are calibrated into strain. • In section 5.2 we review the differential arm length optomechanical plant in the case of detuned signal recycling and arbitrary homodyne angle. We present a reparametrization of this function that is amenable to real-world calibration. • In section 5.3 we examine how systematic errors in the interferometer calibration induce systematic errors in the parameter estimation of compact binary systems. • In section 5.4 we show how to choose calibration line frequencies so as to minimize the uncertainty in the estimated calibration parameters. 5.1 Overview The goal of interferometer calibration is to take the GW readout signal δP(t) (measured in counts, milliamps, milliwatts, or some other kind of hardware unit) and produce an estimate of the freerunning spacetime strain δ h(t) = δL − /L incident along the detector arms. This requires characterizing both the optical dynamics of the interferometer (that is, the transfer function describing how length fluctuation δL − is transduced into power fluctuation δP ) and the feedback control system (figure 5.1), since this system suppresses power fluctuations within the servo bandwidth. Currently, Advanced LIGO uses several methods to calibrate the interferometer readout.51 The “photon calibrator” is the main method used to measure the interferometer’s inverse sensing function, and can be operated while the interferometer is 88 4 ppm 1.4% 3.0% 50% 4 ppm 1.4% NULL 37% SUM Figure 5.1: Simplified diagram of the Advanced LIGO differential arm length sensing and control. The summed photocurrent (“SUM”) is directly proportional to the power fluctuation δP at the interferometer’s antisymmetric port, which is related to the differential arm length fluctuation δL − by a transfer function C( f ) = δP( f )/δL − ( f ). The photocurrent is amplified and fed back to one of the end test masses with a bandwidth ∼ 50 Hz; the open-loop transfer function of this feedback loop is G( f ). Relating the observed power fluctuation δP to the incident freerunning strain requires characterizing the response function [1−G( f )]/C( f ). The differenced photocurrent (“NULL”) is used for diagnostic purposes only. Numbers next to the mirrors give the power transmissivities. (For certain portions of this chapter, the SRM is assumed to be 20 % transmissive.) 89 in its nominal operating state. An auxiliary, amplitude-modulatable 1 μm laser is reflected from one of the end test masses, thereby producing a radiation force. The applied power is measured with a NIST-traceable photodiode, so that the amount of applied force (assuming no in-vacuum clipping or other nonidealities) is known to better than 1 %.129 With a knowledge of the test mass suspension dynamics, this known force can be expressed as an equivalent free-mass displacement. Therefore, when the interferometer is locked, the inverse sensing function [that is, the transfer function from freerunning free-mass displacement (in meters) to GW readout (in milliwatts, milliamps, or digital counts)] can be measured directly. Two other methods are used as cross-checks, and require the interferometer to unlocked. 1. One method method uses a simple Michelson formed by the beamsplitter and the input test masses to calibrate an input test mass actuator against the main laser wavelength. Then a single arm is locked with the main laser, allowing the end test mass actuator to be calibrated against the input test mass actuator. Therefore, the end test mass control signal (in counts) can be calibrated into meters. 2. The other method uses the auxiliary green locking system. A green laser is locked to each arm, producing an rf beat note. A VCO with a known voltageto-frequency actuation strength is locked to this beat note. An end test mass actuator (with a control signal in counts) can then be calibrated against the VCO control signal (in hertz). 5.2 Differential arm length optomechanical plant In chapter 2 we showed that the differential arm length optomechanical plant during the first observing run corresponded to nearly pure resonant sideband extraction, with a small amount of antispring detuning. In this section we explore more fully the effect of optical springs on this optomechanical plant. Buonanno and Chen41 derived input-output relations for a signal-recycled Fabry– Pérot interferometer including radiation pressure. From this one can write down the displacement-to-power transfer function,a up to a constant (see, for example, a This transfer function is fundamentally a force-to-power transfer function. However, for consis- tency with the simple RSE case, the force is referred to an equivalent freerunning displacement via the free-mass compliance 1/M ω2 . Magnitude [mW/pm] 90 101 100 10 O1 ideal: φ = 90◦ , ζ = 90◦ O1 actual: φ = 90.5◦ , ζ = 90◦ Design: φ = 90◦ , ζ = 90◦ Design: φ = 74◦ , ζ = 90◦ Design: φ = 74◦ , ζ = 100◦ −1 10−2 0◦ 101 102 103 101 102 103 Phase −45◦ −90◦ −135◦ −180◦ Frequency [Hz] Figure 5.2: Advanced LIGO differential arm length plant, shown for both O1 (Ts = 37 %, Pbs = 700 W) and for the final design (Ts = 20 %, Pbs = 3500 W). In both cases, 20 mA (= 27 mW) of dc readout light is assumed. Ward130 ): v [( ) ( ) ]u 2iβ 2iβ 2Pbs ω20 ts e 1 − rse cos ϕ cos ζ − 1 + r s e sin ϕ sin ζ u δP t [ ] ∝ , δL − 1 + r 2s e4iβ − 2r s e2iβ cos 2ϕ + (K/2) sin 2ϕ ω2a + ω2 iβ (5.1) where the quantities are defined in appendix D. With some algebra (done in appendix D), this can be transformed to / 1+if z δP / / ) / , = g× ( δL − 1 + i f | p|Q p − f 2 | p|2 − ξ2 f 2 (5.2) with quantities defined as follows. • g is an optical gain (in milliwatts per picometer, or something similar). • z is a zero of the transfer function, given by z = fa × cos(ϕ + ζ) − r s cos(ϕ − ζ) , cos(ϕ + ζ) + r s cos(ϕ − ζ) (5.3) sig n, N S un ing /NS det act u De De sig n, n det o un ing /O2 , al H 1 O1 /O nom 2, ina l Qu O1 ant ity 91 fa 42 Hz 42 Hz 42 Hz 42 Hz Ts 37 % 37 % 20 % 20 % ϕ 90 ° 90.5 ° 90 ° 74 ° ζ 90 ° 90(3) ° 90 ° 100 ° Pbs 700 W 700 W 6250 W 6250 W g 4.4 mW/pm 4.4 mW/pm 9.8 mW/pm 9.8 mW/pm z 365.2 Hz 365.2 Hz 753.7 Hz 396.3 Hz | p| 365.2 Hz 364.2 Hz 753.7 Hz 143.8 Hz Qp 0.5 0.501 0.5 2.422 2 2 ξ 2 2 −4.93 Hz 0 Hz 2 2 2 +63.452 Hz2 0 Hz Table 5.1: Parameters for the Advanced LIGO optomechanical plant. where f a is the arm pole. • p is a complex frequency given by p = fa × so that √ | p| = f a 1 − r s e2iϕ 1 + r s e2iϕ 1 − 2r s cos 2ϕ + r 2s (5.5) 1 + 2r s cos 2ϕ + r 2s and √ Qp = (5.4) , | p| 1 = 2 Re p 2 1 − 2r 2s cos 4ϕ + r 4s 1 − r 2s . (5.6) Note that Q p attains a minimum value of 1/2 when p is real. • ξ2 is the square of the spring frequency: ξ2 = f a2 × 2α r s sin 2ϕ , 1 − 2r s cos 2ϕ + r 2s ) /( and may be positive or negative. Here α = 4Pbs ω0 ω4a ML2 . (5.7) The transfer function in equation 5.2 is not given in a pole-zero representation, since the denominator has not been factorized. Because the denominator is quartic, 92 one can always write down analytic expressions for its roots (that is, the poles of the transfer function), but these expressions are quite complicated. Instead, we will make some general remarks and then give pole-zero representations of this transfer function in some limiting cases. First, we remark on the general features of this transfer function, which has three zeros and four poles. In both the spring and antispring cases, it has two zeros at 0 Hz, a left-handed real zero at the frequency z, and a left-handed complex pole pair with a magnitude roughly equal to the frequency | p|. In the spring case, the final two poles constitute a right-handed complex pair with magnitude roughly equal to ¯ 2 ¯1/2 ¯ξ ¯ . In the antispring case, the final two poles are both real, with one left-handed ¯ ¯1/2 and one right-handed, and the magnitudes are again roughly equal to ¯ξ2 ¯ . Second, we remark that this transfer function reduces to the simple RSE case in the limit that ϕ = π/2. In this case, we have Im p = 0, so α = 0. The denominator then / comprises exactly two real poles at the RSE pole frequency f − = f a ×(1+ r s ) (1− r s ). In the numerator, the zero becomes f − (regardless of the value of ζ), and it therefore cancels one of the poles in the denominator. Therefore, the bracketed portion of /( / ) equation 5.2 becomes 1 1 + i f f − , as expected. Third, we quantitatively analyze the limit that the frequency of the spring is much less than the frequency of the RSE pole. This situation may be encountered if either ϕ−π/2 or PBS is sufficiently small, or if M or L are sufficiently large. In this situation, the denominator can be written approximately as the product of two pole pairs: 1. One pair, located in the left-hand s plane, is complex and produces the highfrequency rolloff in the RSE response. The magnitude and Q of this pair are | p| and Q p , as given above. 2. The second pair produces the spring (or antispring) feature at low frequencies. a) In the spring case (ξ2 > 0), the pole pair is complex and lies in the righthand s plane. The magnitude of this pair can be determined by examining the phase θD ( f ) of the denominator: / / f | p|Q p f | p|Q p / ≃ . tan[θD ( f )] = 1 − f 2 | p |2 − ξ 2 / f 2 1 − ξ 2 / f 2 (5.8) The maximum of the spring feature should occur when θD ( f ) = −π/2; that ¯ ¯1/2 ¯ ¯1/2 is, tan θD = −∞. This occurs when f = ±¯ξ2 ¯ . So ¯ξ2 ¯ is the magnitude of this right-handed pole pair. To find the Q of the pair, we note that when 93 ¯ 2 ¯1/2 ¯ 2 ¯1/2 / f = ¯ξ ¯ , the denominator is (approximately) i¯ξ ¯ | p|Q p . Therefore, | p| Q ξ ≃ ¯ ¯1/2 Q p . ¯ξ2 ¯ (5.9) b) In the antispring case (ξ2 < 0), the pole pair is real, with one pole being right-handed and the other left-handed. Correspondingly, there is a gentle rolloff instead of a sharp resonance feature, and there is no phase loss at low frequencies. In either case, the denominator can be written as )( ( ) f2 if f2 if 2 sgn ξ − ¯ ¯1/2 − ¯ 2 ¯ 1 + − . | p|Q p | p|2 ¯ξ2 ¯ Q ξ ¯ξ ¯ (5.10) The optomechanical plant parameters for various Advanced LIGO configurations are given in table 5.1, and the corresponding transfer functions are shown in figure 5.2. 5.3 Systematic calibration errors In this section we will examine how systematic errors in the interferometer calibration affect astrophysical parameter estimation. General discussion and formalism Suppose we have some frequency-domain detector waveform d( f ) which is known to contain an astrophysical signal—here assumed to be a compact binary coalescence. We want to estimate the astrophysical parameters θ that best correspond to d . This analysis is commonly done in a Bayesian framework,131 where the name of the game is to use Bayes’s theorem arrive at a probability density function (pdf) for θ given the measured waveform. Bayes’s theorem says p(θ| d) = p(d |θ) p(θ) , p(d) (5.11) with the quantities defined as follows. 1. p(θ) is the prior pdf for θ (that is, our initial guess at a pdf of θ). 2. p(d |θ) is the probability of observing the waveform d given a certain value of θ. This conditional pdf is determined by the underlying physics that is assumed 94 to generate d from θ (in our case, a combination of analytic and numerical relativity). Any quantity proportional to p(d |θ) is referred to as a likelihood function. 3. p(d) is a normalizing constant referred to as the evidence. 4. p(θ| d) is the posterior pdf for θ (that is, our updated pdf for θ in light of d ). If the detector noise in each frequency bin is Gaussian, we can assume the following for the logarithm of the likelihood function: ¯ ] 1 ℓ ≡ ln p d( f ; λ) ¯ θ ∝ − 2 [ ¯2 ∫∞ ¯¯ h( f ; θ) − d( f ; λ)¯ df . S nn ( f ; λ) (5.12) 0 Here h( f ; θ) is the relativistic waveform expected from a system with parameters θ, and S nn ( f ) is the noise PSD of the detector. d( f ) is (again) the waveform from the detector. For both d and S nn we have now explicitly written the dependence on λ, which refers to the calibration parameters used to estimate the freerunning strain from the detector error signal e( f ). Estimating d from e requires estimating a transfer function referred to as the interferometer’s response function, R : d( f ; λ) = R( f ; λ) e( f ) ¯ ¯2 S nn ( f ; λ) = ¯R( f ; λ)¯ S ee ( f ). (5.13) (5.14) We now want to know how the maximum of ℓ shifts if small calibration errors are introduced. Suppose e( f ) is generated from a GW waveform h( f ; θtrue ) with parameters θtrue . Suppose also the interferometer’s calibration parameters are λtrue , but we’ve in/ stead made a slightly wrong estimate λ. Then d( f ) = h( f ; λtrue ) × R( f ; λ) R( f ; λtrue ), ¯ ¯2 / true) and S nn ( f ) = S (nn ( f ) × ¯R( f ; λ) R( f ; λtrue )¯ . Therefore, ℓ(θ, λ) ≡ ln p(d |θ, λ) ¯2 ¯ ¯ ∫∞ ¯¯ ¯ h( f ; θtrue )¯ 1 R( f ; λtrue ) h( f ; θ) ¯¯2 ¯ × ¯1 − ∝− df , true) 2 R( f ; λ) h( f ; θtrue ) ¯ S (nn (f ) (5.15) 0 where we now explicitly indicate the functional dependence on λ as well as θ. For brevity, we’ll find it convenient to define r(θ, λ) = R( f ; λtrue ) h( f ; θ) − h( f ; θtrue ). R( f ; λ) (5.16) 95 We want to know how the maximum of ℓ changes in the presence of nonzero ∆λ = λ − λtrue , and how this maps to an equivalent ∆θ = θ − θtrue . With a little calculus, it is straightforward to show       ∆θ1 ∆λ1 ℓθ1 θ1 ℓθ1 θ2 · · · ℓθ1 θM ℓθ1 λ1 ℓθ1 λ2 · · · ℓθ1 λN        ℓθ θ   ∆θ2   ℓθ λ   ∆λ2  · · · · · · ℓ ℓ ℓ ℓ θ θ θ θ θ λ θ λ 2 1 2 2 2 2 1 2 2 2 M  N       . = .  .  , (5.17)  . .. ..  .. ..  .. ..    ..   .   . . . . . .  .   . . .  .    ∆θ M ∆λ N ℓθM θ1 ℓθM θ2 · · · ℓθM θM ℓθM λ1 ℓθM λ2 · · · ℓθM λN | {z } | {z } ≡H ≡M where ℓµ i µ j is the second partial derivative of ℓ with respect to parameters µ i and µ j: 1 ∂2 ℓ = =− ℓµ i µ j 2 ∂µ i ∂µ j ∫∞ 0 ] ∂2 r ∗ ∂r ∗ ∂r × 2 Re r+ . true) ∂µ i ∂µ j ∂µ i ∂µ j S (nn (f ) df [ (5.18) In equation 5.17 we have made note of the fact that the matrix H on the left-hand side is the Hessian of ℓ with respect to θ. Additionally, to the matrix on the righthand side we have assigned the letter M. This allows us to write the relationship between calibration errors ∆λ and parameter estimation errors ∆θ as ∆θ = J ∆λ with J = H−1 M. (5.19) The letter J is chosen to remind the reader of a Jacobian matrix, since this equation expresses how first-order changes in λ are related to first-order changes in θ. Application to 2 pN parameter estimation For the purpose of this simulation, we want to examine the effect of calibration errors on a simple Bayesian parameter estimation using stationary-phase 2 pN waveforms (see Röver et al.132 for an example of this kind of parameter estimation). In the 2 pN approximation, the parameters comprising θ are • the total mass M = M1 + M2 , • the symmetric mass ratio η = M1 M2 /(M1 + M2 )2 , • the coalescence time t c , • the coalescence phase ϕc , and • the luminosity distance D , which is degenerate with the inclination angle of the source. 96 The 2 pN stationary-phase waveform (expressed as a frequency-domain strain) is then ( ) { [ ]} (GM)5/6 5η 1/2 −7/6 h( f ; θ) = 2/3 3/2 f × exp − i 2π f t c + ϕc + ψ2pN ( f ; M, η) , (5.20) D 6 2π c / where ψ2pN is a polynomial in (πGM f )1/3 c, with coefficients involving η.132 1 For the calibration model, we assume λ comprises six parameters ( g, z, | p|, Q p , ξ2 , and a), and the response function is given by R( f ; λ) = 1 − A( f ; a) D( f ). C( f ; g, z, | p|,Q p , ξ2 ) (5.21) Here C is the interferometer sensing function, with parameters given by equation 5.2. A is the suspension transfer function. In this simulation, the ultimate (electrostatic) and penultimate (magnetic) actuators are used, with a crossover around 20 Hz. The strength of the penultimate actuator is assumed to be constant for the duration of the run, while the strength a of the ultimate actuator (in newtons per volt) is assumed to drift because of charge accumulated on or near the test mass. D is a set of (known) analog and filters connecting the error signal (in milliamps) to the control signal (in volts). The resulting open-loop transfer function has a UGF of 180 Hz, which is required to stabilize the right-handed poles of the spring feature; this OLTF is shown in figure 5.3. The total response function R = 1/S is shown in figure 5.4. The true astrophysical parameters were chosen to be M = 65 M⊙ , η = 0.247, t c = 0.250 ms, ϕc = 0.60 rad, and D = 410 Mpc. The true calibration parameters were chosen to be the detuned RSE NS/NS-optimized configuration given in table 5.1 We compute J in equation 5.19 using an automatic differentiation routine.b This yields the following relationship between systematic errors in the calibration parameters and systematic errors in the astrophysical parameters:    ∆ M/M −80       ∆η/η 177   / 1      ∆ t c (1 ms)  = 3  260    ∆ϕ /(1 rad) 10  −2174  c   ∆D/D −180 b ad in python. −21 −80 −64 −51 40 198 136 94 851 −1689 812 857 −819 −1218 −1913 −1751 −71 329 3 194   ∆ g/g   −129   ∆ z/z    274    ∆| p|/| p|    . 570   ∆Q p /Q p    −3148    ∆ξ2 /ξ2    961 ∆a/a (5.22) 97 Magnitude [m/m] 101 100 10 −1 Open loop Closed loop Phase [deg.] 101 180 135 90 45 0 −45 −90 −135 −180 101 102 102 Frequency [Hz] 103 103 Figure 5.3: Open- and closed-loop transfer functions of the differential arm length loop used in the simulations in this chapter corresponding to NS/NS-optimized detuned RSE. Compared to O1, the UGF has been increased by a factor of 3, the time delay of the loop has been reduced, and additional digital shaping is applied to compensate the complex RSE feature around 150 Hz. We can use this matrix to set astrophysically motivated requirements on our calibration. For example, if we wish to determine the system’s total mass to better than 1 %, then we should aim for no more than 2.4 % fractional systematic error on each of the calibration parameters, since 2.4 % × (80 + 21 + 80 + 64 + 51 + 129) × 10−3 = 1.0 %. Application to tests of general relativity The above analysis can be used to examine the impact of calibration errors on GWbased tests of general relativity (GR). These tests often involve augmenting the waveform model with adding additional parameters that should be equal to zero in GR.126,133 One example is the use of GW merger data to bound the mass m (or equivalently the Compton wavelength λ = h/mc) of the graviton.134 Phase [deg.] Magnitude [pm/mW] 98 102 101 100 10−1 10−2 180 135 90 45 0 −45 −90 −135 −180 101 102 101 102 Frequency [Hz] 103 103 Figure 5.4: Response function 1/S for the 125 W detuned RSE scheme considered here. A massive graviton is subject to the dispersion relation E 2 = m2 c4 + p2 c2 , which implies that a gravitational wave with wavelength λ propagates with a speed given by ( ) v2 p 2 c 2 hc 2 = ≃ 1− . λE c2 E2 (5.23) For a GW inspiral, this has the effect of introducing a 1 pN phase term ϕg ( f ) to the inspiral waveform: ϕg ( f ) = − πD c B , ≡ − f λ2 (1 + z) f (5.24) where D is a distance-like quantity (not necessarily equal to the luminosity distance D ) defined by Will,134 and z is the source redshift. Note that B has the dimensions of a frequency. In standard GR, m = 0, so λ = ∞ and hence B = 0 Hz. Analysis of the GW150914 signal set a limit λ > 1013 km at 90 % confidence.126 99 Repeating the above analysis with the B parameter included yields the following matrix equation:  ∆ M/M   −2392       ∆η/η 3563    /     ∆ t c (1 ms)   2723 = 1   /   3 ∆ϕc (1 rad) 10  −10 380     ∆D/D   −414    / ∆B (1 kHz) 2259 280 −6577 −1003 −401 9713 1512 530 5231 1812 250 −24 280 −5246 −40 −327 −92 −294 6349 918 1111 −2769  ∆ g/g      4140    ∆ z/z     ∆| p|/| p|  −380 3382  .    2373 −12 518  ∆Q p /Q p     2 2  312 694    ∆ξ /ξ  ∆a/a −1135 2580 −1607 (5.25) In the standard GR case B = 0 (that is, λ = ∞), a systematic error ∆B will result in a reportedly finite wavelength λ according to equation 5.24. For D/(1 + z) ∼ 400 Mpc and ∆B ∼ 100 Hz (a typical value expected from equation 5.25 assuming a fewpercent systematic error in the calibration parameters), the resulting Compton wavelength is λ ∼ 1013 km. Note that this is of the same order as the upper limit placed on GW150914. This indicates that unmodeled systematic calibration errors at the few-percent level could already produce a fictitious finite value for the graviton Compton wavelength, at least in the detuned interferometer configuration discussed here. 5.4 Optimal transfer function measurements from the Fisher matrix Many optical, mechanical, and optomechanical systems are often assumed to be linear and time-invariant. Under this assumption, the parameters of such a system can be estimated by measuring the system’s transfer function. Although one can always measure a transfer function with a standard swept-sine routine, there are several situations that lead us to think more carefully about our measurement strategy. 1. In the case of systems such as alignment control systems, seismic isolation systems, and mechanical suspensions, transfer function measurement can be quite expensive, as it can require exciting the system down to millihertz frequencies. 2. In the case of important data channels such as the differential arm length readout of Advanced LIGO, transfer function measurement is expensive because such measurements necessarily invalidate certain parts of the data. 100 This is particularly true for the continuous transfer function measurement required to track time-varying changes in the interferometer calibration (the “calibration lines”); data at the line frequencies must be thrown out. With a little forethought, one can choose the measurements so as to maximize the amount of information learned about the system’s parameters. One method for quantifying the amount of information gained is to compute the Fisher matrix of the measurement.135 The inverse of the Fisher matrix provides a lower bound (the so-called Cramér–Rao bound) on the covariance matrix of the estimated parameters. Good, straightforward explanations of the Fisher matrix and the Cramér–Rao bound are few and far between. Two such resources are David Wittman’s “Fisher Matrix for Beginners”136 and the technical appendix to the Dark Energy Task Force report.137 General discussion We define the following notation: we have a linear, time-invariant (LTI) system whose transfer function H( f ) depends on a certain set of parameters θ = (θ1 , θ2 , . . . , θ M ). We can probe this system by exciting its input with a signal x( f ) and reading back the response y( f ) = H( f )x( f ) + n( f ), where n( f ) is some readout noise. Our goal is to produce an estimate Ĥ(θ; f ) given our observed response y( f ), our excitation x( f ), and our estimate of the noise n( f ). In practice, we excite the system with sinusoids at frequencies f 1 , f 2 , . . . , f N . We record the complex excitation amplitudes x1 , x2 , . . . , x N and the response amplitudes y1 , y2 , . . . , yN . These amplitudes have been corrupted by noise whose amplitudes are n 1 , n 2 , . . . , n N ; in general, we have yα = Hα xα + n α ; α = 1, 2, . . . , N, (5.26) where Hα = H( f α ). On the other hand, given an estimate Ĥ(θ; f ) of the system, we can write down a set of estimated amplitudes ŷ1 , ŷ2 , . . . , ŷN , with ŷα = Ĥα (θ)xα ; α = 1, 2, . . . , N, (5.27) Our goal is to find the value of θ which makes the estimated responses { ŷα } approach the observed responses { yα }. To this end, we can write down a likelihood function L(θ) ∝ p({ yα }|θ), where p({ yα }|θ) is the probability of having observed the 101 amplitudes { yα } given a certain value of θ. From here on we will assume that the noise is Gaussian, which results in a likelihood functionc [ ] N∑ −1 | y − ŷ (θ)|2 α α L(θ) ∝ exp − . 2| n α |2 α=0 (5.28) How should we place our N frequencies so as to maximize the amount of information we can learn about H ? Intuitively, we know we should choose our frequencies so as to maximize the curvature of L (or, equivalently, the curvature of ln L) with respect to θ. To find an expression for the curvature, we vary θ and keep track of terms up to second order: ¯ ¯ ∑ ∂[ln L] ¯¯ ∑ ∑ ∂2 [ln L] ¯¯ ¯ 1 ln L(θ + δθ) ≃ ln L¯θ + ¯ δθ i + ¯ δθ i δθ j . 2 i j ∂θ i ∂θ j ¯ ∂θ i ¯ i θ (5.29) θ Once we have found parameters θ0 which maximize ln L, the first-derivative terms will vanish, leaving only the second-derivative (that is, curvature) terms. These curvature terms are the elements of the Fisher matrix F:d ¯ ∂2 [ln L] ¯¯ Fi j = − ¯ ∂θ i ∂θ j θ0 [ ]¯ ∑ | yα − ŷα (θ)|2 ¯ ∂2 ¯ = ¯ 2| n α |2 ∂θ i ∂θ j α θ0 ]¯ [ ∗ ∑ 1 ¯ ( ) ∂2 ŷα ∂ ŷα ∂ ŷα ∗ ∗ ¯ . = − y − ŷ + cc α α ¯ 2 ∂θ ∂θ ∂θ i ∂θ j i j α 2| n α | θ0 If our estimate ŷ is unbiased, we expect yα − ŷα → 0, and thus [ ∗ ]¯ ∑ 1 ∂ ŷα ∂ ŷα ¯¯ Fi j = Re . 2 ∂θ i ∂θ j ¯θ0 α |nα | (5.30a) (5.30b) (5.30c) (5.31) Since ŷα = Ĥα xα , we have [ ∂Ĥα∗ ∂Ĥα Re Fi j = 2 ∂θ i ∂θ j α |nα | ∑ | xα | 2 ]¯ ¯ ¯ ¯ , ¯ (5.32) θ0 c If one is not comfortable with likelihood functions, note that − ln L is equivalent to the usual χ2 statistic used for curve fitting. d Note that F = −H, where H is the Hessian of ln L evaluated at θ . Strictly speaking F is the 0 observed Fisher information, which is to be contrasted with the expected Fisher information. See Efron and Hinkley138 for more information. 102 which is the discrete-frequency equivalent of the expression found by L. Price.135 Following the usual convention, we’ll write σ(Hα ) = | n α /xα |.e The inverse of the Fisher matrix provides a lower bound for the covariance matrix: Σ ≥ F− 1 , (5.34) where the inequality is understood to be elementwise. This is the so-called Cramér– Rao bound. Single-frequency excitations As an example, we apply the above concepts to the problem of estimating the parameters of a single-pole system H( f ) = g/(1 + i f /p) using an excitation at only one frequency. This system describes, for example, the optical response of a resonant Fabry–Pérot cavity to length or frequency perturbations. In this context, g is called the optical gain and p is called the cavity pole. For concreteness we can consider the Advanced LIGO differential arm length plant, for which (during O1) we have g ≃ 3.2 mA/pm and p ≃ 350 Hz. For this system, our parameter vector is θ = (g, p), so the Fisher matrix will be 2 × 2. As a start, we’ll consider the case where we have only a single excitation at a frequency f 1 . This results in only a single observation y1 .f In this instance, the Fisher matrix is  p2  2 p + f 12 1   F= 2  2 σ1  g p f 1 ( )2 p2 + f 12 ( ( g p f 12  )2   ,   ) 2 2 p2 + f 12 g2 f 12 (5.35) p2 + f 1 and as a result, the covariance matrix Σ is bounded elementwise from below by F− 1 :  ( 2 )  2 2 ) p + f 1 ( 2 2 1 2   − 3   4 p + f1 g p 2p ( . ( ) ) Σ ≥ σ1  3 2 p2 + f 12  p2 + f 12   − 3 2 2 2 gp g p f1 (5.36) ( i) (r) α = yα + iyα , and e Alternatively, we can decompose each observation into real/imaginary parts: y likewise for Hα and n α . Then we can write down an alternative Fisher matrix Fi j = N ∑ ∑ (β) (β) | xα |2 ∂Ĥα ∂Ĥα . ¯ (β) ¯2 ∂θ i ∂θ j α=1 β∈{r,i} ¯ n α ¯ In the case n(αr) = n(αi) , this reduces to the definition given above. f However, since y is complex, it contains two pieces of information: y = y(r) + iy(i) . 1 1 1 1 (5.33) 103 ¯ ¯ ¯Σ( k, k)/ k2 ¯1/2 ¯ ¯ ¯Σ( k, p)/ k p¯1/2 ¯ ¯ ¯Σ( p, p)/ p2 ¯1/2 ¯ ¯ ¯(det Σ)/ k2 p2 ¯1/4 Magnitude / σ 102 101 100 10−1 101 102 103 Excitation frequency [Hz] Figure 5.5: Normalized Cramér–Rao bounds on the covariance matrix Σ for a singlefrequency transfer function estimate of the system H( f ) = g/(1 + i f /p), with g = 3.2 mA/pm and p = 350 Hz. Here we have assumed a white readout noise and a flat excitation amplitude. From here, the goal is to choose f 1 so as to provide the “optimal” Σ. To make progress, we need to make assumptions about σ1 , which means making assumptions about the excitation amplitude x1 and the readout noise amplitudes n 1 . The next sections explore two simple cases for x1 and n 1 . Flat excitation and white readout noise To start with, we’ll assume that the readout noise n α is Gaussian and white as a function of frequency. (In the case of the differential arm length in Advanced LIGO, the assumption of whiteness is true only above 100 Hz or so.) We’ll also assume that the excitation amplitude is flat: xα = x0 for all α. Then σ1 = σ( f 1 ) ≡ σ (that is, it is independent of frequency). In this case, the elements of Σ are minimized as follows: 1. To minimize Σ(g, g), one should choose f 1 = 0 Hz. This results in ] [ σ2 −σ2 p/g , Σ(0) ≥ −σ2 p/g ∞ (5.37) which is evidently unacceptable for simultaneous estimation of g and p. 104 Magnitude × ρ ¯ ¯ ¯Σ( k, k)/ k2 ¯1/2 ¯ ¯ ¯Σ( k, p)/ k p¯1/2 ¯ ¯ ¯Σ( p, p)/ p2 ¯1/2 ¯ ¯ ¯(det Σ)/ k2 p2 ¯1/4 101 100 101 102 103 Excitation frequency [Hz] Figure 5.6: Normalized Cramér–Rao bounds on the covariance matrix Σ for a single-frequency transfer function estimate of the system H( f ) = g/(1 + i f /p), with g = 3.2 mA/pm and p = 350 Hz. Here we have assumed that the response y has a constant SNR ρ above the readout noise n. 2. To minimize Σ(g, p), one should again choose f 1 = 0 Hz. p 3. To minimize Σ(p, p), one should choose f 1 = p/ 2. This results in [ ] ( p ) 9 σ2 −σ2 p/g Σ p/ 2 ≥ . 4 −σ2 p/g 3σ2 p2 /g2 (5.38) p 4. To minimize det Σ, one should choose f 1 = p/ 3. This results in [ ] ( p ) 16 σ2 −σ2 p/g Σ p/ 3 ≥ . 9 −σ2 p/g 4σ2 p2 /g2 (5.39) In figure 5.5 we plot the elements of Σ, along with det Σ, as a function of f 1 , assuming that the Cramér–Rao bound is saturated. Constant-SNR excitation Often we can do better than a flat excitation amplitude. If we already have reasonable knowledge of g, p, and n, it is desirable to aim for the measurement to have a constant SNR. For our purposes, we define this as ρα = | yα /n α | = | xα Hα /n α |. There(r,i) fore, we choose | xα | = ρ| n α |/| Hα |, where ρ is our target SNR, and hence σα = | Hα |/ρ. 105 With this choice of excitation amplitude, the Fisher matrix is   f 12 1 ( )   g2 g p p2 + f 12  2  F=ρ  2 2  f f   1 ( 1 ) ( ) g p p2 + f 12 p2 p2 + f 12 and the covariance matrix satisfies   2 ) ) g ( 2 g( 2 2 2 p + f1 − p + f1  2 1  p p   Σ≥ 2  g ( ) ( 2 ) . 1 2 2 2 2 ρ − p + f1 p + f1 p f 12 (5.40) (5.41) As expected, the bound on the covariance goes down like ρ2 . To minimize the bounds on both Σ(g, g) and Σ(g, p), one should again choose f 1 = 0 Hz. However, this again results in Σ(p, p) = ∞: [ Σ(0) ≥ g2 /ρ2 − g p/ρ2 − g p/ρ2 ∞ ] . (5.42) This time, the bounds on both Σ(p, p) and det Σ are minimized by choosing f 1 = p: [ ] g2 /ρ2 − g p/ρ2 Σ(p) ≥ 2 . (5.43) − g p/ρ2 2p2 /ρ2 In figure 5.6 we plot the elements of Σ, along with det Σ, as a function of f 1 , assuming that the Cramér–Rao bound is saturated. Application to differential arm length calibration For NS/NS-optimized detuned operation at full power We want to apply the above formalism to the problem of estimating the parameters of the differential arm length loop using photon radiation pressure. A known radiation force F( f ) = − M ω2 x( f ) is applied to the test mass, resulting in a power fluctuation P( f ) at the dark port. x( f ) and P( f ) are related via the reciprocal response function S( f ) = 1/R( f ), whose parameters g, z, | p|, Q p , ξ2 , and a we would like to constrain. We assume the calibration will consist of five lines placed somewhere between 7 Hz and 3 kHz, each with the same, fixed SNR ρ. The Fisher matrix in this case is ( ) ∗ 5 ∑ Ŝ 1 Ŝ ∂ ∂ α α F i j = ρ2 , ¯ ¯2 Re ∂λ i ∂λ j α=1 ¯Ŝ α ¯ (5.44) 1. 8 1. 75 8 1. 90 9 1. 05 92 1. 0 9 2. 8 0 2. 4 10 2. 2 2. 3 2. 4 2. 5 2. 5 2. 55 5 2. 60 5 2. 65 57 0 2. 7 3. 5 0 3. 0 25 log f 5 2 2log f4 log f 3 log f 2 2. 3. 3. . 5 . 5 2. 5 2. 5 2 2 2 2 1. 2. 2. 55 60 65 70 .2 .3 .4 .5 98 04 10 75 00 25 106 log f 1 log f 2 log f 3 log f 4 log f 5 Figure 5.7: Posterior distribution of optimal line frequencies for differential arm length optical plant estimation, as determined by maximizing the volume of the Fisher information of the measurement. The differential arm length optical plant assumed here is the 125 W NS/NS-optimized plant described in table 5.1, with a loop shown in figure 5.3. ( )⊺ with λ = g z | p| Q p ξ2 a . The Fisher matrix determinant det F is maxi- mized with respect to the five line frequencies via MCMC. In the MCMC, it is important to enforce f 1 ≤ f 2 ≤ f 3 ≤ f 4 ≤ f 5 in order to produce well-localized posteriors on the frequencies. The posterior pdf of the frequencies is shown in figure 5.7. The maximum a posteriori frequencies are 82, 133, 145, 359, and 2847 Hz. The Fisher matrix (evaluated at the maximum a posteriori frequencies) is inverted to yield a bound on the covariance matrix Σ. The resulting relative covariance ma- 107 Magnitude 1.04 1.02 1.00 0.98 0.96 101 102 103 2. 0◦ Phase 1. 0◦ 0. 0◦ −1.0◦ −2.0◦ 101 102 Frequency [Hz] 103 Figure 5.8: 300 randomly generated realizations of statistical errors in the parameters g, z, | p|, Q p , ξ2 , and a for the response function 1/S corresponding to the NS/NSoptimized, detuned, full-power operation described in the text. The green diamonds indicate the chosen calibration line frequencies. The realizations are drawn according to the covariance matrix given in equation 5.45 with an SNR of ρ = 100. The blue dashed lines give the envelope of the 3σ uncertainties associated with these errors. This shows that the frequency-dependent statistical error of the five-line sensing scheme considered here is better than 3 % in magnitude and 2 ° in phase in the GW band. trix Σ(rel) (with elements Σ(rel) = Σ i j /λ i λ j ) is ij  1.122     0.8552  1.052     2 2 2   0.387 1 −0.556 −0.203 (rel) . Σ = 2  ρ  −1.172 −0.9882 0.6062 1.492    2 2  −1.022 −0.5352 0.6562  1.04 1.40   2 2 2 2 2 2 0.533 0.341 −0.424 −0.781 −0.62 0.658 (5.45) 108 Since none of these values is significantly larger than unity, if the line SNRs are chosen to be ρ = 100 then the parameter uncertainties and covariances can be constrained to about 1 % or better. In figure 5.8 we show the expected frequencydependent statistical uncertainty in the interferometer response function, with the errors drawn from a distribution given by the optimal covariance matrix written above with ρ = 100. Evidently, the Cramér–Rao bound implies that we can achieve better than 3 % uncertainty in magnitude and 2 ° in phase with this scheme. For nearly pure RSE operation at lower power We repeat the above analysis for an O2-like configuration with 50 W of input power, no significant homodyne angle offset, and an 8 Hz antispring. Explicitly, g = 6.0 mW/pm, z = 365.2 Hz, | p| = 364.2 Hz, Q p = 0.501, ξ2 = −8.0 Hz2 , and a = 2.2 × 10−7 N/V. Be- cause z and | p| are nearly equal, and Q p is nearly 1/2, attempting to estimate all six parameters will result in excessively large uncertainties. Therefore, we assume that z and Q p are known and fixed, and we include only g, | p|, ξ2 , and a in the Fisher matrix. For O2, a sensible “by-hand” choice of lines is 8, 37, 330, and 1080 Hz. This results in a relative covariance matrix   2 0.886     2 2 − 0.625 0.602 1   Σ(rel) = 2  . 2 2  ρ −0.6512 0.461 1.29   −0.1022 −0.02252 0.1022 0.2912 (5.46) Figure 5.9 shows the resulting frequency-dependent errors in the response function. On the other hand, optimizing the four calibration line frequencies to maximize the determinant of the Fisher matrix yields lines at 10.4, 49, 199, and 2807 Hz. The resulting relative covariance matrix is   0.6522    −0.4472 2 0.513 1   Σ(rel) = 2  . 2 2  ρ −0.5382 0.357 1.25   −0.1982 −0.06912 0.2462 0.2742 (5.47) Figure 5.10 shows the resulting frequency-dependent errors in the response function. 109 Magnitude 1.04 1.02 1.00 0.98 0.96 101 102 103 2. 0◦ Phase 1. 0◦ 0. 0◦ −1.0◦ −2.0◦ 101 102 Frequency [Hz] 103 Figure 5.9: Response function errors (analogous to figure 5.8) for four calibration line frequencies picked by hand for an O2-like configuration. The hand-picked calibration lines result in a Cramér–Rao bound det Σ ≥ 2.6 × 109 for ρ = 100, while the optimized calibration lines have a bound det Σ ≥ 1.0 × 109 . By comparing figures 5.9 and 5.10, one can see that this does correspond to an improvement in the response function uncertainty; however, the improvement is modest: the maximal error of 3.6 % and 1.6 ° is reduced to 3.0 % and 1.4 °. 110 Magnitude 1.04 1.02 1.00 0.98 0.96 101 102 103 2. 0◦ Phase 1. 0◦ 0. 0◦ −1.0◦ −2.0◦ 101 102 Frequency [Hz] 103 Figure 5.10: Response function errors (analogous to figure 5.8) for four optimized calibration line frequencies for an O2-like configuration. Compared to the handpicked frequencies in figure 5.9, the optimized frequency choice improves the response function magnitude uncertainty around 200 Hz and improves the phase uncertainty between 30 and 500 Hz. 111 6 Searching for dark matter with laser interferometers This chapter considers the possible use of laser interferometers to detect large macroscopic dark matter objects interacting gravitationally with the interferometer test masses. Numerical simulations are presented showing detection rates for both a Newtonian gravitational coupling and a Yukawa coupling. The work presented here also appears in a preprint by Hall et al.139 6.1 Previous work Laser interferometers have already received some consideration as dark matter detectors, either because of possible GW signatures of dark matter, or because of the possibility of local dark matter interaction with the interferometer itself. In terms of GW signatures of dark matter, Bird et al.140 argue that some of the universe’s dark matter density may consist of primordial black holes (PBHs) with masses between 20 M⊙ and 100 M⊙ , and that GW150914 could have been generated from a binary system of two such black holes. A PBH scenario was also considered by Sasaki et al.141 Brown142 and Hanna143 conducted searches in Initial LIGO data for sub-solar-mass binary systems; these searches are motivated in part by the possibility of binary MACHOs, which may have component masses from 0.15 M⊙ to 0.9 M⊙ based on mi- crolensing surveys. In terms of non-GW tests of dark matter using laser interferometers, Stadnik and Flambaum144 consider the case of a dark matter scalar field which couples quadratically to the electromagnetic field in a laser interferometer. Additionally, Seto and Cooray (whose results we will quote below) considered direct Newtonian pertur- 112 bations of LISA test masses from PBH flybys, with masses of order 1014 kg ( ∼ 10−17 M⊙ ).145 Rather than a primordial black hole or an aggregation of baryonic matter, a macroscopic dark matter object could be a gravitationally bound aggregate of nonbaryonic particles. Such macroscopic objects can form from the gravitational infall of dark matter particles into primordial overdensities, and these objects can subsequently be broken apart by tidal disruption.146 6.2 Newtonian and Yukawa-type interactions In this chapter we consider the possibility of Yukawa gravitational interactions between standard and dark matter particles, with a potential Vi j (r) = − ] Gm i m j [ 1 + (−1)s δ i δ j e−r/λ , r (6.1) where i and j may each refer to either standard matter (SM) or dark matter (DM). m i and m j are the masses of the particles. If the interaction is mediated by a scalar, then s = 0; if it is mediated by a vector, then s = 1. δSM and δDM are Yukawa couplings for SM and DM, and λ is a Yukawa screening length. For the problem of detecting a DM object with a test mass made of SM, we are particularly interested in the SM–DM coupling g = δSM δDM . Note that this Yukawa interaction reduces to the usual Newtonian interaction for either δSM = 0 or δDM = 0. This potential leads to a central force F i j (r) = − ∂Vi j ∂r r̂ = − ] ( Gm i m j [ r) s − r/λ r̂. ( − 1) δ δ e 1 + 1 + SM DM λ r2 (6.2) Torsion balance tests of the equivalence principle by, for example, Schlamminger et al.147 set an upper limit of 10−3 for δSM , assuming λ > 10 m. A limit on δDM is set by the self-interaction cross section σDD of dark matter, which is constrained by structure formation to be σDD /m ≲ 0.1 m2 /kg for v/c ≃ 103 . This leads to a constraint ( ) 1 kg 1/4 δDM ≲ 5 × 10 × . m DM 9 (6.3) Here and below we assume that the radius r DM of the dark matter objects are much less than the interferometer arm length, and that the Yukawa screening length is much less than n1/3 , where n DM is the number density of dark matter objects in DM the universe. 113 6.3 Detection with laser interferometers Analytical Newtonian case Seto and Cooray145 considered the Newtonian interaction of primordial-mass black holes (1013 to 1017 kg) with test masses of a LISA-like detector (arm length L = 5 × 106 km), and derived the expected event rate given the measured local dark matter density in the galaxy (ρ = 0.011 M⊙ /pc3 = 7 × 10−22 kg/m3 ). We will briefly review their analysis and results before moving on to the new numerical analysis performed in this work. They performed an analytical analysis in which each black hole (with mass M ) flies at normal incidence through the plane of the detector, with a fixed speed v = 350 km/s. If the distance of closest approach to a particular test mass is R , then the acceleration time series of the test mass is a(t) = [ GMR R 2 + (vt)2 ]3/2 , (6.4) which in the frequency domain is ) ( 2GM 2π f R a( f ) = K1 , Rv v (6.5) where K 1 is the first modified Bessel function of the second kind. With the frequencydomain acceleration in hand, the optimal matched-filter SNR ϱ can be computed via ¯2 ∫∞ ¯¯ a( f )¯ ϱ = 4 df , S aa ( f ) 2 (6.6) 0 where S aa ( f ) is the PSD of the detector’s acceleration noise. Seto and Coray assume −15 a flat acceleration noise of S 1/2 m s−2 Hz−1/2 , so that ϱ can be computed aa = 3 × 10 analytically in terms of M , R , and L.a To compute the expected rate η̇ (in events per year) of events above a certain critical SNR ϱ∗ , Seto and Cooray assume that the entire dark matter halo consists of black holes of a particular mass M . If R ∗ is the radius which yields an SNR of ϱ∗ , then a The derivation and rates quoted here implicitly assume R < L, since the acceleration of only one test mass is considered; Seto and Cooray refer to this as the “close-approach” regime. They additionally give a derivation for the “tidal” regime in which R > L and the accelerations of all test masses must be considered. 114 Acceleration [m/s2 ] 2 ×10−15 1 0 −1 −2 −200 −150 −100 −50 0 50 Time [ms] 100 150 200 Figure 6.1: Example acceleration time series for dark matter interactions with Advanced LIGO, here shown for 1000 kg objects. the number of flybys per unit time with SNR ϱ ≥ ϱ∗ is πR ∗2 F ≡ η̇(ϱ∗ ), where F = ρv/M is the event flux. Seto and Cooray derive the following formula for this rate: ) )3/5 ( v ρ η̇(ϱ) = 0.01 yr M ϱ 350 km/s 7 × 10−22 kg/m3 ) ( )4/5 ( 3 × 10−15 m s−2 Hz−1/2 L , 5 × 106 km S 1/2 aa ( −1 ) ( ) ( ) 1 × 1014 kg 1/5 5 4/5 ( (6.7) suggesting that the prospects for primordial black hole detection with LISA are quite marginal, with less than one high-SNR event expected even after 10 years of observation. Numerical Newtonian analysis We performed a Monte Carlo simulation of the interaction of compact objects (such as primordial mass black holes) with laser interferometer test masses, both for Advanced LIGO and for LISA. In this simulation, a dark matter object is incident on the plane of the detector with a random impact parameter and random angles of incidence. The speed is chosen from a distribution that takes into account both the zero-mean, randomly oriented thermal velocity distribution of the dark matter in the halo as well as the nonzero, 115   ASD of acceleration m s−2 Hz−1/2 10−14 10−15 10−16 10−17 10−18 10−19 10−20 101 102 Frequency [Hz] Figure 6.2: Example acceleration spectra for dark matter interactions with Advanced LIGO, corresponding to the time series in figure 6.1. The dashed line shows the assumed acceleration noise of Advanced LIGO for the purposes of the simulations in this chapter. nonrandom velocity of the solar system through the halo. This distribution is a displaced Maxwell–Boltzmann distribution (that is, the three-dimensional analog of a Rice distribution), which we now derive. The joint distribution for the individual components ẋ, ẏ, and ż of the clump velocities is [ ] ( ẋ⊙ − ẋ)2 + ( ẏ⊙ − ẏ)2 + ( ż⊙ − ż)2 p( ẋ, ẏ, ż) = ( , )3/2 exp − 2σ2 2πσ2 1 where v⊙ = √ (6.8) 2 2 ẋ⊙ + ẏ⊙2 + ż⊙ = 220 km/s is the speed of the solar system, and σ = 270 km/s is the thermal velocity spread of the clumps in the halo. The corresponding speed distribution p(v) can be found by changing variables into spherical coordinates ( ẋ = v cos ϕ sin θ, ẏ = v sin ϕ sin θ, and ż = v cos θ), and assuming (without loss of gen- 116 103 102 Cumulative rate [yr−1 ] 101 100 10−1 10−2 10−3 10 aLIGO, 0.1 kg aLIGO, 10 kg aLIGO, 1000 kg −4 10−5 LISA, 109 kg LISA, 1011 kg 10−6 10−7 LISA, 1013 kg 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 SNR Figure 6.3: Cumulative event rate η̇(ϱ) for the number of dark matter events per year with SNR of at least ϱ, assuming a Newtonian interaction. erality) that the solar system velocity lies entirely in the z direction. Then ∫∞ ∫∞ ∫∞ 1 = d ẋ d ẏ d ż p( ẋ, ẏ, ż) −∞ ∫2π = −∞ ∫π (6.9a) −∞ ∫∞ [ 2 ] ẋ + ẏ2 + (v⊙ − ż)2 1 dϕ dθ dv v2 sin θ ( exp − ) 2 3/2 2σ2 2 πσ 0 0 0 ( 2 ) ∫∞ ( )1/2 (v v) v⊙ + v 2 2 v ⊙ = dv , exp − sinh 2 π σ v⊙ 2σ σ2 | {z } 0 (6.9b) (6.9c) ≡ p(v) where the integrand in the last line is the expression for the speed distribution p(v). Example acceleration time series for 10 kg objects interaction with an Advanced LIGO detector are shown in figure 6.1, and the corresponding spectra are shown in figure 6.2. Additionally, in this simulation the effect of the flyby on all test masses is accounted for when computing the differential acceleration, and the matched-filter SNR is computed using the full, frequency-dependent acceleration noise PSD. 117 M = 0.1 kg   Rate with SNR > 1 yr−1 102 M = 10 kg 102 102 M = 1000 kg λ = 102 m λ = 103 m λ = 104 m 101 101 101 100 100 100 10−1 10−1 10−1 10−2 10−2 10−2 10−3 10−3 10−3 10−4 10−4 10−4 107 106 105 g 104 103 102 101 107 106 105 104 103 102 101 107 106 105 104 103 102 101 g g Figure 6.4: Cumulative event rate η̇(1) in Advanced LIGO for the number of dark matter events per year with SNR of at least 1, as a function of Yukawa coupling g and screening length λ. M = 109 kg 2   Rate with SNR > 1 yr−1 10 M = 1011 kg 2 10 10 101 101 101 100 100 100 10−1 10−1 10−1 10−2 10−2 10−2 λ = 107 m λ = 108 m λ = 109 m 104 103 102 101 100 104 g 103 102 101 100 104 103 102 101 100 g M = 1013 kg 2 g Figure 6.5: Cumulative event rate η̇(1) in LISA for the number of dark matter events per year with SNR of at least 1, as a function of Yukawa coupling g and screening length λ. For each detector and each DM object mass, several thousand events are simulated, and the histogram of SNRs is recorded. From this, the cumulative event rate η̇(ϱ) is computed. The results are shown in figure 6.3. Consistent with the Seto and Cooray analysis, the prospects for DM detection with LISA are not promising. Further, the prospects for DM detection with Advanced LIGO are even less promising. Numerical Yukawa analysis The numerical Yukawa analysis is similar to the numerical Newtonian analysis, with the Newtonian force law replaced with the Yukawa force law given in equa- 118 tion 6.2. For each value of the DM mass, the simulation is run for several values of coupling g and screening length λ, and the unity-SNR event rate η̇(1) is recorded. The results are shown in figure 6.4 for Advanced LIGO and 6.5 for LISA. In the lightest-mass case considered ( M = 0.1 kg), detection rates become promising for g ≳ 104 , with a screening length λ ∼ 1 km. Shapiro delay One effect not considered in the above analysis and simulation is the Shapiro delay in the interferometer arms. Shapiro delay in laser interferometers has already been analyzed, for example, in the context of intentional modulation for Fabry– Pérot cavities by Ballmer et al.148 and in the context of asteroid flybys for LISA by Chauvineau et al.149 In the simple case of a stationary mass M located a distance d along an arm of length L at an impact parameter b, the round-trip time delay is150 ∆t = − 2GM 4d(L − d) ln . c3 b2 (6.10) For a given mass and impact parameter, the delay is maximized for d = L/2, giving ( ) ( ) ∆ t = − 4GM/c3 ln L/b . For Advanced LIGO, a 1000 kg mass located 1 m from the midpoint of an arm (corresponding to a characteristic interaction frequency v/b ∼ 300 Hz) would produce a peak displacement c∆ t of about 10−23 m, which is too small to be detected. The logarithmic dependence of the time delay on b means that a peak displacement of 1 × 10−21 m (which might be detectable) would require an impact parameter that is smaller than the Planck length. 6.4 Search strategy A significant practical impediment to this kind of search in Advanced LIGO is the lack of colocated interferometers. The screening lengths considered here (λ ≲ 10 km) are much smaller than the current (and planned) baseline separations of advanced laser interferometers (typically thousands of kilometers). Therefore, a putative DM interaction event would register in only one interferometer at a time. However, a search could be conducted over data from the H1 and H2 initial LIGO interferometers. Additionally, some third-generation interferometers such as the Einstein Telescope151 and LISA6 are planned to have colocated interferometers. 119 7 Future work In this chapter we give a few suggestions for how to improve the current Advanced LIGO detectors and what to think about for future detectors. 7.1 Interferometery Suggestions for sensitivity improvements have been given by Miller et al.152 Some possible minor alterations to the interferometer configuration or sensing are as follows: • The use of paired in-vacuum length sensors to facilitate out-of-loop and crosscorrelation noise estimates. An out-of-loop reflection sensor would be particularly useful, since it is difficult to estimate the residual frequency noise by other methods. • The inclusion of pick-off sensors for the signal recycling cavity. • Greater jitter suppression before the laser is injected into the vacuum system. Pointing noise from the injection-locked oscillator greatly contaminated the interferometer’s differential sensitivity. While one can attempt to mitigate the jitter coupling through careful thermal tuning and alignment offset adjustment, it is much less painful to mitigate the jitter itself by using a higher-finesse pre-modecleaner, or two pre-modecleaners in series. • Active optical stabilization of the rf sideband amplitudes. Residual amplitude modulation (RAM) has the potential to introduce extra sensing noise into length and angular control loops, including dc error point offsets.153 Active RAM suppression could be achieved via heterodyned electrical and thermal EOM feedback154 or direct rf feedback to the EOM.155 A more drastic configuration change may be made by removing the signal-recycling mirror in order to alleviate some technical issues associated with the presence of 120 10−13 Arm reflectivity imbalance Arm pole imbalance Schnupp asymmetry Radiation pressure Total 10−14 37% Magnitude [m/Hz] 10−15 10 SR M No S RM −16 10−17 10−18 10−19 10−20 101 102 103 Frequency [Hz] Figure 7.1: Expected frequency noise coupling into differential arm length for the case of a 37 % signal recycling mirror and for the case of no signal recycling mirror. In each case the input test mass transmissivities have been adjusted as described in the text to give a differential arm pole f − ≃ 350 Hz. The assumed input power is 25 W. The calculation is done according to equation 29 in Izumi et al.21 The assumed arm reflectivity imbalance is δ r a = 0.0029, the arm pole imbalance is δ f a = 0.24 Hz, the differential reduced mass is δµ = 5 g, and the differential arm amplitude gain is δ g a = −0.036. 121 10−10 % 37 10−11 SR Magnitude [m/RAN] M 10−12 N o SR M Carrier Arm reflectivity imbalance Arm pole imbalance Schnupp asymmetry Radiation pressure 45 MHz transmission Total 10−13 10−14 10−15 10−16 10−17 10−18 101 102 103 Frequency [Hz] Figure 7.2: Expected intensity noise coupling into differential arm length for the case of a 37 % signal recycling mirror and for the case of no signal recycling mirror. In each case the input test mass transmissivities have been adjusted as described in the text to give a differential arm pole f − ≃ 350 Hz. The assumed input power is 25 W. The calculation is done according to equation 26 in Izumi et al.21 122 a signal recycling cavity. Here we consider a scenario in which the interferometer is operated at O1 power (25 W) with no signal recycling mirror. To maintain the differential arm pole at f − ≃ 350 Hz, the input test mass transmissivity can be set to Ti = 11.5 %, giving an arm finesse F = 52. Some consequences of this optical configuration change are as follows. • Sensing and control issues. Eliminating the SRM simplifies the interferometer sensing-and-control scheme by removes the signal-recycling length degree of freedom and the two angular degrees of freedom of the mirror. • Frequency and intensity noise couplings. Removing the SRM (and subsequently altering the input test mass transmissivities) makes the interferometer less susceptible to frequency noise, but more susceptible to intensity noise (figures 7.1 and 7.2). However, the changes in susceptibility are modest (about a factor of 5 at most). • Modematching. With the SRM, the signal-recycling cavity must be modematched to the arms, and the output modecleaner must be matched to the signal-recycling cavity. Eliminating the SRM simplifies this problem, requiring only that the output modecleaner be matched to the arms. This reduces the potential for modematching-induced output port losses, which degrade the shot-noise-limited SNR and limit the achievable amount of squeezing. • Power-recycling losses. With the lower arm finesse, the circulating power in the interferometer becomes more sensitive to losses in the power-recycling cavity. To ensure the circulating arm power drops by no more than a few percent of its zero-loss value, a PRC loss on the order of 10−4 or less is required (equation 2.17b). • Michelson motion. The coupling Ξm = π/2F of Michelson motion into differential arm length readout increases by roughly a factor of 10. Since the shot noise of the Michelson sensor can be subtracted from the GW readout signal with feedforward, this means that the Michelson displacement noise must be kept sufficiently small. Looking further into the future, we can ask how the frequency coupling transfer function could look for a 40 km interferometer of the type considered for the so-called Cosmic Explorer.156 We consider a dual-recycled Fabry–Pérot Michelson with input test mass transmissivity Ti = 7 %, PRM transmissivity Tp = 0.5 %, SRM transmissivity Ts = 10 %, a circulating arm power Pa = 2 MW, test masses weighing 320 kg, and a Schnupp asymmetry LM = 0.8 m. The frequency-to-displacement coupling transfer function is shown in figure 7.3. For the chosen parameters, the 123 10−13 Arm reflectivity imbalance Arm pole imbalance Schnupp asymmetry Radiation pressure Total 10−14 Magnitude [m/Hz] 10−15 10−16 10−17 10−18 10−19 10−20 101 102 103 Frequency [Hz] Figure 7.3: Expected frequency noise coupling into differential arm length for a possible 40 km detector as described in the text. The calculation is again done according to equation 29 in Izumi et al.21 The various imbalances are the same as the imbalances assumed for the Advanced LIGO coupling transfer functions described above. The magnitude of the coupling for the 40 km detector is similar to the magnitude of the Advanced LIGO coupling. If the displacement noise of the 40 km detector is similar to the displacement noise of Advanced LIGO, this indicates that the laser frequency noise requirement need not be significantly more stringent than the current Advanced LIGO requirement. 124 transfer function is well approximated by (10−15 m/Hz) × (100 Hz/ f ). If the sensitivity of the Cosmic Explorer is about 8 × 10−21 m/Hz1/2 between 20 Hz and 1 kHz, then the required frequency noise at the interferometer input must be less than ( ) ( / ) 1 μHz/Hz1/2 × 100 Hz f in order to contribute negligibly to the total displacement sensitivity. In fact this is not so different from the requirements for Advanced LIGO.35 However, the controls problem must also be considered. For a 40 km in- terferometer the interferometer’s response to frequency fluctuations (using PDH reflection locking) will have infinitely deep notches at positive integer multiples of the arm FSR c/2L = 3.7 kHz. This restricts the bandwidth of the common-mode frequency stabilization loop to less than a few kilohertz, and hence restricts the amount of suppression that can be achieved. However, if a macroscopic differential arm length offset L− is incorporated into the interferometer topology, the FSR notches acquire a finite depth. Either L− could be made large enough to make the notches negligibly deep, or the first few notches could be electronically compensated to produce a stable servo loop. 7.2 Calibration Some possible future directions in interferometer calibration are as follows: • Repeating the analysis given in chapter 5 using waveforms that include merger and ringdown signals (for example, those described by Ajith et al.157 ). • Investigating other methods of optimizing calibration line placement. In chapter 5 we minimized the volume of the covariance of the interferometer’s optomechanical plant parameters. However, in the end we are actually interested in minimizing the volume of the covariance of the astrophysical parameters (mass, spin, and so on) of a particular signal. To this end, we may want to minimize a different quantity, such as the uncertainty in the interferometer response function in a particular frequency band. Additionally, the line optimization technique described in the calibration chapter could be put to work more generally as a way to optimally extract information about the dynamics of LTI systems. For example, this technique could be applied to optical plant measurement for angular control loops, where the timescale of the dynamics (seconds or tens of seconds) typically require long transfer function measurements. 125 A Preliminary notions about loops This appendix presents a basic overview of feedback loops. Deeper discussion can be found in various controls textbooks and review articles, such as the article by Bechhoefer.158 A.1 Introduction Let P( f ) denote the transfer function of some linear, time-invariant system (the plant) which we want to control. The plant takes some input x( f ) and produces some output y( f ) = P( f ) x( f ). We want to take y (in this context called the error signal), pass it through some filter K( f ) (the compensator) to produce a control signal k( f ), and feed this control signal back into the input of P( f ). The resulting servo loop is characterized by its open-loop transfer function (OLTF) H( f ) = K( f ) P( f ). The OLTF indicates to what extent (and at what frequencies) the feedback system can suppress external fluctuations. If an external fluctuation x( f ) is injected into the loop, then after one trip around the loop it will arrive just before the injection point as a fluctuation b( f ) = H( f )x( f )/[1 − H( f )]. Just after the excitation point, the fluctuation is a( f ) = b( f ) + x( f ) = x( f )/[1 − H( f )]. We call H/(1 − H) the system’s closed-loop transfer function and denote it as H . We call 1/(1 − H) the system’s loop suppression function and denote it as H .a For the feedback system to be stable, an external perturbation x( f ) must not produce an infinite in-loop perturbation a( f ); that is, we must have a/x = 1/(1 − H) ̸= ∞, or H( f ) ̸= +1. a Sometimes 1/(1 − H) is referred to as the closed-loop transfer function.159 (A.1) r n x k 126 e P K a u b Figure A.1: Anatomy of a generalized servo loop, consisting of a plant P and a compensator K , which together form an open-loop transfer function H = PK . Any external noise x injected into the loop causes a fluctuation in the error signal e. The error signal is fed through K to produce a control signal k, which is summed back into the loop in order to suppress the external noise, so that its contribution to the residual loop noise r is x/(1 − H). In general, the equipment used to construct the servo loop will also introduce sensing noise n, which appears in the residual loop noise as Hn/(1 − H). Also depicted is a loop excitation point u; the OLTF H can be measured by monitoring the signals a and b just after and just before this excitation. ¯ (¯ ) Said another way, if the OLTF has unity gain ¯ H( f )¯ = 1 , the phase must not be ( ) zero arg H( f ) ̸= 0◦ .b ¯ ¯ Any frequency f 0 for which ¯ H( f 0 )¯ = 1 is called a unity-gain frequency (UGF) for H , and arg H( f 0 ) is called the phase margin at that particular UGF. A small phase margin causes the magnitude of the loop suppression function H to exceed 1 around the UGF, meaning that the loop amplifies noise rather than suppressing it; this is called gain peaking. Additionally, at any frequency f 1 for which arg H( f 1 ) = 0◦ , the ¯ ¯ quantity ¯ H( f 1 )¯ is called the gain margin at that particular frequency. b Warning: there exist two popular sign conventions for loop analysis, depending on how the control signal k is fed back into the loop. This appendix is written with the convention that k is summed into the loop, as depicted in figure A.1. If instead one uses the convention that k is differenced into the loop, then a number of statements made in this appendix must be modified. Among other things, loop instability occurs when H = −1 (that is, | H | = 1 and arg H = ±180◦ ), and the expressions for the closed-loop transfer function and loop suppression function are H = H/(1 + H) and H = 1/(1 + H). 127 A.2 Crossovers Sometimes, a feedback loop employs multiple LTI blocks in parallel. Such loops are common in systems which rely on multiple actuators to provide feedback. For example, when frequency-locking a solid-state laser such as an NPRO to an optical cavity, feedback to the length of the laser crystal can be applied using a PZT to strain the crystal, or a heater to control the thermal expansion of the crystal. The former provides several megahertz of actuation range with a bandwidth of a few tens of kilohertz, and the latter provides several gigahertz of actuation range with a bandwidth of a few hertz. For robust, long-lived locks, both actuators are needed. Consider a loop with OLTF H = H1 + H2 (i.e., two blocks H1 and H2 are added in parallel). The usual loop stability condition requires H( f ) = H1 ( f ) + H2 ( f ) ̸= +1; i.e., H1 /(1 − H2 ) ̸= +1. We define Ξ ≡ H1 /(1 − H2 ) (A.2) to be the crossover transfer function of H1 and H2 ; it characterizes the relative ¯ ¯ strength of H1 versus H2 at various frequencies. Any frequency f 0 for which ¯Ξ( f 0 )¯ = 1 is a crossover frequency between H1 and H2 .c A.3 Measurements We now turn to some topics in how to measure transfer functions. Again let P( f ) denote some LTI system, with an input and output. The canonical measurement of the transfer function P( f ) is done by injecting random noise x( f ) into the output, which will generate a signal y( f ) = P( f )x( f ) at the output. The estimate P̂( f ) of P( f ) is made by computing160 P̂( f ) = S x y( f ) S xx ( f ) (A.3) , where S x y ( f ) is the (complex) cross-spectral density of x and y, and S xx ( f ) is the (real) auto-spectral density (or power spectral density) of x. Equivalently, we will write the cross-spectral density as 〈 x∗ y〉, as this notation is more suggestive of how spectral densities are estimated in practice. A common figure of merit for the quality of a particular measurement is the coherence γ2x y ( f ) = ¯ ¯ ¯S x y ( f )¯2 S xx ( f )S yy ( f ) , c Sometimes the crossover transfer function is instead defined as H /H .159 1 2 (A.4) 128 which can attain any value from 0 (no correlation between x and y) to 1 (perfect correlation between x and y). We consider a loop whose OLTF H( f ) we would like to measure. In order to do this, we inject some excitation u into the loop, and we measure just after (a) and just before ( b) the excitation (see figure A.1). As mentioned above, these test points are given (in the frequency domain) by u 1−H Hu b= . 1−H a= (A.5) (A.6) With a, b, and u in hand, one can see that there are (at least) four ways to extract the OLTF. 1. Compute an estimate of the CLTF ∗ 〈 u b〉 ˆ H= ∗ 〈 u u〉 (A.7) and then do algebra to compute Ĥ . 2. Compute an estimate of the loop suppression function Ĥ = 〈 u∗ a〉 〈 u∗ u〉 (A.8) and then do algebra to compute Ĥ . 3. Compute an estimate of the OLTF directly using a and b only Ĥ = 〈a∗ b〉 . 〈a∗ a〉 (A.9) This is the technique used with network analyzers (such as DTT161 ) when making Fourier-based (for example, broadband) transfer function measurements. 4. Compute an estimate of the OLTF directly using a, b, and u: Ĥ ′ = 〈 u∗ b〉 . 〈 u ∗ a〉 (A.10) This is the technique used with network analyzers when making demodulationbased (for example, swept-sine) transfer function measurements. The last of two these is often the most convenient: the estimated OLTF can be read directly off a network analyzer, with no algebra required. However, we will now 129 show that the third estimator ( Ĥ ) has the drawback of being biased, while the other three do not. All realistic loops have some noise x which is also injected into (or generated within) the system under feedback control. In the presence of both noise x and excitation u, the test points for our measurement are u+x 1−H Hu + x b= , 1−H a= (A.11) (A.12) and so, assuming that x and u are uncorrelated, the direct OTLF estimate is Ĥ = 〈a∗ b〉 H 〈 u∗ u〉 + 〈 x∗ x〉 = . 〈a∗ a〉 〈 u∗ u〉 + 〈 x∗ x〉 (A.13) This equation is telling us that attempting to directly estimate H in this fashion will succeed only if the excitation u is much stronger than the other noise x in the loop. If | u| ≫ | x|, then Ĥ → H , as desired. But if | u| ≪ | x|, then Ĥ → 1, regardless of the true value of H . In other words, Ĥ is a biased estimator of H . Moreover, examining the coherence of the test points may be misleading, since in the limit u → 0, a and b are both perfectly correlated with x. Then γ2ab → 1. ˆ One can show that H , Ĥ , and Ĥ ′ are all unbiased estimators, converging respec- tively to H , H , and H for any u ̸= 0, regardless of the relative strength of u and x.d In situations where it is hard to satisfy | u| ≫ | x| (for example, angular loops which cannot tolerate much excitation beyond the quiescent noise level of the loop), it may be advantageous to use these alternative estimators if possible. A.4 Noise Let us again focus on the example of locking a laser to an optical cavity by monitoring the phase fluctuation ϕ( f ) of the light in reflection. The phase fluctuation may arise either from length fluctuation L( f ) of the cavity, or from frequency fluctuation ν( f ) of the laser light: ϕ( f ) = (2π/c)[ν(0) L( f ) + ν( f ) L(0)]. (A.14) The former noise is an example of a displacement noise — a noise in the error signal arising from actual motion of the system of interest. The latter noise is an d Of course, if | u| ≪ | x| by many orders of magnitude, extracting a good transfer function estimate may be impractical. 130 example of a sensing noise — a noise in the error signal arising from whatever is used to sense the system of interest. A closely related concept has to do with what kinds of noise limit the performance of a servo loop. Consider again loop with OLTF H . Suppose the loop has a certain amount of displacement noise x( f ), as well as some sensing noise n( f ) injected after the displacement noise. Now suppose we were able to measure the true residual r( f ) between x and n. We would find r= x Hn + . 1−H 1−H (A.15) This first term on the right-hand side of this equation says that as the gain | H | is increased, the loop’s displacement noise is more aggressively suppressed. The second term says that when | H | > 1, the sensing noise is impressed onto the residual. If the first term dominates the residual, the loop is said to be gain limited, and the residual r within the loop bandwidth can be reduced by increasing the loop gain. If the second term dominates the residual, the loop is said to be sensing-noise lim- ited, and the residual r within the loop bandwidth cannot be reduced by increasing the loop gain. Note that the loop’s error signal e( f ) in general does not accurately represent the true in-loop residual, since e= x n + ̸= r. 1−H 1−H (A.16) In other words, continuing to increase the loop gain | H | will make the error signal smaller, even if the loop’s actual residual is sensing-noise limited. This is the motivation for placing an “out-of-loop” sensor at nearly the same place as the in-loop sensor; this out-of-loop sensor will see the true in-loop residual in combination with the sensor’s own sensing noise.e e However, the out-of-loop sensor will not see sensing noises common to both sensors. 131 B Median estimation of amplitude spectral densities Bin-by-bin median averaging of amplitude spectral densities (ASDs) is sometimes preferred over traditional rms averaging for its increased robustness against transients.162 In this note we examine some of the subtleties of median averaging techniques, with a particular focus on Gaussian noise. The central result of this appendix—that in the limit of large sample size, the median is biased downward from the mean by a factor of (ln 2)1/2 —is already known in the gravitational-wave data analysis community.163–165 Here we give a simple demonstration of this fact using the aLIGO H1 OMC null stream. B.1 Problem statement Given a time series h(t), how does one arrive an an estimate of the ASD S hh ( f )1/2 ? Spectrum analyzers, DTT, Matlab’s pwelch, etc., usually produce an ASD estimate using some flavor of the Welch method:166 1. Split h(t) into N segments, each of length T . The segments can be chosen to overlap by some factor (often 50 %). 2. Apply a windowing function w(t) to each segment h i (t) and compute the Fourier transform H i ( f ) of w(t)h i (t). 3. At each Fourier frequency f , compute the rms of the magnitudes of the Fourier transforms (and then apply a normalization factor C) in order to arrive at an estimate of the ASD: ( 1/2 Ŝ hh ( f ) −1 ¯ ¯ 1 N∑ ¯ H i ( f )¯2 =C N i=0 )1/2 . (B.1) 132 Note that rms averaging of the ASDs amounts to mean averaging of the PSDs, so this step is often referred to as “mean averaging”. B.2 Gauss and Rayleigh distributions When one says that h(t) consists of Gaussian noise, one means that in each frequency bin, the quadrature components x = Re H( f ) and y = Im H( f ) of the noise are independent, identically distributed, Gaussian random variables with zero mean and variance σ2 . That is, their joint pdf is ( 2 ) 1 x + y2 p(x, y) = p(x)p(y) = exp − . 2πσ2 2σ2 (B.2) When estimating the ASD, we are interested not in x or y individually, but rather the quantity r = (x2 + y2 )1/2 . (We often discard the phase information ϕ = arctan y/x.) This quantity does not follow a Gaussian distribution, but rather a Rayleigh distribution:160;a;b ( ) r r2 p(r) = 2 exp − 2 σ 2σ for r ≥ 0, (B.3) and p(r) = 0 otherwise. For this distribution, we will be interested in the following quantities: • the mode M = max p(r) = σ; r • the mean ∫∞ µ= r p(r) dr = ( π )1/2 2 σ ≃ 1.25σ; (B.4) (B.5) 0 • the rms ∞ 1/2 ∫ p ψ =  r 2 p(r) dr  = 2σ ≃ 1.41σ; (B.6) 0 and a Explicitly: ¯ writing ¯x = r cos ϕ and y = r sin ϕ, the pdf p(x, y) transforms to p(r cos ϕ, r sin ϕ) ¯det J(r, ϕ)¯, where J(r, ϕ) is the Jacobian of the coordinate transformation. Integrating over ϕ gives the Rayleigh distribution p(r). b If one recasts this analysis in terms of PSDs rather than ASDs, then the quantity s = r 2 follows an exponential distribution, with p(s) = exp(− s/ζ)/ζ and ζ = 2σ2 .163,164 133 • the median m, which is defined as follows: 1 = 2 ∫m 0 ⇒ ( ) m2 p(r) dr = 1 − exp − 2 2σ m = (ln 4)1/2 σ ≃ 1.18σ. (B.7) (B.8) For the Rayleigh distribution, these four quantities are distinct, with σ < m < µ < ψ.c B.3 rms versus median averaging The usual algorithm for estimating an averaged ASD from multiple data segments is to compute the rms: that is, for each bin, we compute a quantity proportional ( )1/2 / N . Therefore, this estimate r is an estimate of ψ, as to r = r 20 + r 21 + · · · + r 2N −1 defined above (and hence we can write this estimate as ψ̂). Conversely, the median { } estimate m̂ = med r 0 , r 1 , . . . , r N −1 converges to m.d Therefore, in each frequency bin of a Gaussian-noise ASD, the median value is biased downward from the corresponding rms value by a factor b ∞ = m/ψ = (ln 2)1/2 ≃ 0.83, in the limit of an infinite number of non-overlapping segments. The papers by Abbott et al.163 and Allen et al.164 discuss the value of the bias when the two foregoing assumptions are not satisfied. For a finite number of nonoverlapping segments, the sample median is not equal to the true median of the underlying Rayleigh distribution. Instead, the bias is given by [ ] N (−1) n+1 1/2  ∑    if N is odd,  n n =1 b(N) =      b(N − 1) if N is even. (B.10) As one might anticipate, b(N) → (ln 2)1/2 as N → ∞. Table B.1 gives the value of b for several values of N . c We note in passing that in the case of Gaussian noise superimposed on a coherent line—for example, shot noise superimposed on the vibrational mode from some optic—the ASD will instead follow a Rice distribution: ( 2 ) ( rν ) r + ν2 r I , (B.9) p(r) = 2 exp − 0 σ 2σ2 σ2 where I 0 is the modified Bessel function of the first kind, and ν describes the height of the line in the ASD. d Note that since the function x 7→ x2 is strictly increasing for x ≥ 0, one does not need to square the ASDs before computing the median. 134 N b 1 1.0000 3 0.9128 10 0.8635 30 0.8427 100 0.8356 300 0.8336 1000 0.8328 ∞ 0.8326 Table B.1: Median bias factor b (to four decimal places) for N non-overlapping segments. In the case of overlapping segments, the above formula for b is still not correct, since the data in adjacent segments are correlated (i.e., one does not truly have N independent samples from the underlying distribution). If median averaging with overlap is desired (and the overlap is ≤ 50 %), Allen et al. suggest computing the medians m and m′ of the k even-numbered and k′ odd-numbered segments individually, correcting by the appropriate bias factors b and b′ , and then taking the weighted rms of the two values: [ k(m/b)2 + k′ (m′ /b′ )2 ψ̂ = N ]1/2 . (B.11) This prescription is referred to as mean–median averaging. B.4 Real-world example We now demonstrate some of the above claims using data from the Advanced LIGO H1 null stream. The Advanced LIGO instruments read out the light at the antisymmetric port using two photodiodes (labeled A and B), each with photocurrent I A (t) and I B (t). The sum I + = I A + I B is the error signal for the differential arm length. The difference I − = I A − I B is the null stream. Since A and B are balanced, the null stream should contain only those noises which are uncorrelated between the two photodiodes.e e Of course, it is only approximately true that A and B are balanced. In practice, the balancing of A and B is tuned digitally in order to minimize the appearance of correlated noise in the null stream. For H1 the required digital balancing is 0.3 %. 135 Figure B.1: Three ASD estimates of the H1 OMC null stream. The first is the usual bin-by-bin rms averaging of the 720 individual ASDs. The second is the bin-by-bin median estimate, divided by (ln 2)1/2 . The third is the bin-by-bin median estimate, without this correction factor incorporated. 45 Rayleigh distribution Mode: 5.61 × 10−8 mA/Hz1/2 Mean: 7.03 × 10−8 mA/Hz1/2 Median: 6.61 × 10−8 mA/Hz1/2 rms: 7.93 × 10−8 mA/Hz1/2 Histogram of values 40 Number observed 35 30 25 20 15 10 5 0 0. 0 0. 5 1. 0 1. 5 2. 0 Value of ASD [mA/Hz1/2 ] 2. 5 ×10−7 Figure B.2: Histogram of the H1 OMC null stream ASD at f = 100 Hz, assembled from N = 720 individual segments. The observed median m was used to estimate the parameter σ for the underlying Rayleigh distribution. In particular, the null stream should be dominated by (1) the shot noise of the light from the interferometer, and (2) the electronics noise of the photodiodes and 136 their readout chains. In both cases we can reasonably expect that the noise in each quadrature of each frequency bin is Gaussian. We have taken two hours of raw H1 null stream data ( f samp = 16 384 Hz)f and processed it as follows using gwpy: 1. We split the data record into N = 720 segments (with 50 % overlap and T = 10 s). 2. We apply a Hann window to each segment, and then apply a discrete Fourier transform. 3. We take the magnitude of each DFT and normalize it, thereby producing 720 individual ASD estimates (with units of mA/Hz1/2 ). (The time stream I − is already calibrated into milliamps.) 4. We produce averaged ASD estimates, either by rms averaging, or median averaging. No rebinning is performed. 5. We undo the magnitude of the digital decimation filter, which otherwise produces significant rippling of the spectrum above 1 kHz. Figure B.1 shows the rms and median ASD estimates for these two hours of null stream data. The median is plotted twice, both with and without the (ln 2)1/2 correction. Only with the correction factor applied does the median estimate coincide with the rms estimate for this Gaussian channel. Figure B.2 shows a histogram of the 720 ASD estimates at f = 100 Hz. We compute the mean, median, and rms values, and use the median to estimate the parameter σ (i.e., the mode) of the underlying Rayleigh distribution. The Rayleigh distribution is then plotted on top of the histogram. f More specifically, the data record is H1:OMC-DCPD_NULL_OUT_DQ, from 00:00:00Z to 02:00:00Z 2015–06–07. 137 C Optical cavities In this appendix we will review some concepts related to optical cavities and Pound– Drever–Hall locking. C.1 Definitions We consider a Fabry–Pérot interferometer of length L, and with optics of reflectivity r a and r b . The circulating field E circ , the reflected field E refl , and the transmitted field E tr are given by E circ (ω) = E 0 (ω) E refl (ω) = E 0 (ω) E tr (ω) = E 0 (ω) ta (C.1) 1 − r a r b e2iωL/c − r a + r b e2iωL/c (C.2) 1 − r a r b e2iωL/c t a t b eiωL/c 1 − r a r b e2iωL/c . (C.3) By examining (C.1), we see that the frequency response is characterized by periodic resonance: when e2iωL/c = 1, the denominator becomes very small since r a r b ≃ 1 for reasonably reflective mirrors. We now define some commonly-used quantities relating to Fabry–Pérot cavities. The periodicity of the frequency response is called the free spectral range (FSR): f FSR = c/2L. (C.4) The width of the resonance is characterized by the cavity pole f p = ωp /2π, which is found by setting the denominator of (C.1) to zero and solving for ω. The result is18,37 2πi f p = − ¯ c ¯¯ ln r a r b ¯; 2L (C.5) 138 in other words, it is a single, real pole on the left-hand side of the s-plane. To derive the next three quantities, we examine the circulating power / ¯ ¯ t2a (1 − r a r b )2 t2a Pcirc (ω) ¯¯ E circ (ω) ¯¯2 ( ). =¯ =¯ ¯ = 4r a r b ¯1 − r a r b e2iωL/c ¯2 P0 (ω) E 0 (ω) ¯ 2 ωL 1+ sin c (1 − r a r b )2 (C.6) The ratio E circ (0)/E 0 (0) is the cavity amplitude gain ta , 1 − ra rb g= (C.7) and the cavity power gain is G = g2 . The frequency for which Pcirc /P0 = 1/2 is the half-width half-max frequency: ] [ c 1 1 − ra rb f HWHM = . × arcsin 2L π 2(r a r b )1/2 (C.8) In combination with the FSR, we can now define the finesse F= f FSR = 2 f HWHM [ arcsin π/2 ]. 1 − ra rb (C.9) 2(r a r b )1/2 In the high-finesse limit, we have f HWHM ≃ f p and F≃ π(r a r b )1/2 ≃ (C.10) 2π , (C.11) 1 − ra rb L where L is the round-trip loss in the cavity (including transmission loss). This last √ √ approximation follows by writing r a = 1 − La and r b = 1 − Lb , with La , Lb ≪ 1, and L = La + Lb . Finally, some words about coupling. If r a < r b , the cavity is said to be overcoupled; if r a = r b the cavity is said to be critically coupled, and if r a > r b , the cavity is said to be undercoupled. In the high-finesse limit, G and F have the following relationships in the strongly overcoupled ( r b = 1) and critically coupled cases: G ≃ 2F/π (strongly overcoupled) (C.12a) G ≃ F /π (critically coupled). (C.12b) Another important quantity is the visibility: v = 1− Prefl (0) . Prefl ( f FSR /2) (C.13) 139 C.2 Pound–Drever–Hall locking In the Pound–Drever–Hall technique,167 the light into the cavity is phase-modulated with an index Γ at an rf frequency Ω/2π, and the rf beatnote is read out in reflection of the cavity. We assume the cavity is on resonance at dc, and is subject to a phase modulation of index γ at audio-band frequency ω/2π. The field into the cavity is p [ ] E in = 2E 0 cos ω0 t + Γ cos Ω t + γ cos ω t E0 = p eiω0 t eiΓ cos Ω t eiγ cos ω t + cc 2 { E 0 iω0 t iΓ iΓ iγ iγ ≃p e 1 + eiΩ t + e−iΩ t + eiω t + e−iω t 2 2 2 2 2 } [ Γγ i(Ω+ω)t i(Ω−ω)t i(−Ω−ω)t i(−Ω+ω)t ] − e +e +e +e + cc, 4 (C.14a) (C.14b) (C.14c) where we have kept terms of order 1, Γ, γ, and Γγ. Upon reflection from the cavity, { E 0 iω 0 t iΓ iΓ iγ iγ E refl = p e r 0 + r Ω eiΩ t + r −Ω e−iΩ t + r ω eiω t + r −ω e−iω t 2 2 2 2 2 ]} Γγ [ i(Ω+ω)t i(Ω−ω)t i(−Ω−ω)t i(−Ω+ω)t r Ω+ω e + r Ω −ω e + r −Ω−ω e + r −Ω+ω e + cc. − 4 (C.15) We have made the assumption that the carrier is exactly resonant, so r 0 is real. We further assume the rf sidebands are exactly antiresonant, so r Ω is real and equal to r −Ω . This means additionally that r −ω = r ∗ω and r Ω−ω = r ∗Ω+ω . Then { E0 E refl = p eiω0 t r 0 + iΓ r Ω cos Ω t + iγ Re(r ω ) cos ω t − iγ Im(r ω ) sin ω t 2 } − Γγ cos(Ω t) [Re(r Ω+ω ) cos ω t − Im(r Ω+ω ) sin ω t] + cc (C.16) E0 (C.17) ≡ p eiω0 t ψ + cc, 2 where ψ refers to the terms in curly brackets. Then the detected optical power is Prefl = E 2refl ]2 E 20 [ iω t −iω0 t ∗ 0 e ψ+e ψ = 2 P0 = × 2ψ∗ ψ + (terms at 2ω0 ) 2 = P 0 ψ∗ ψ. (C.18a) (C.18b) (C.18c) (C.18d) 140 The PDH error signal is made by demodulating the reflected power Prefl at the rf frequency Ω/2π. If we are interested at the error signal response at audio frequency ω/2π, then when computing Prefl we therefore need only to consider terms that oscillate at ±Ω ± ω. Then { Prefl = 2P0 Γγ cos(Ω t) [− r 0 Re(r Ω+ω ) + r Ω Re(r ω )] cos ω t } + [r 0 Im(r Ω+ω ) − r Ω Im(r ω )] sin ω t + (other terms). (C.19a) Demodulating at Ω/2π and low-pass filtering yields the following phase-to-power transfer function: ) ( 〈Prefl 〉Ω = P0 Γ − r 0 r ∗Ω+ω + r Ω r ∗ω . γ (C.20) We can recast this as a frequency-to-power transfer function by recalling the Laplacedomain relation 2πν(s) = γ̇(s) = sγ(s). Additionally, in the more general case that Γ is not much smaller than 1, we can replace Γ by 2J0 (Γ)J1 (Γ). Therefore, the PDH reflection transfer function is D( f ) = P( f ) − r(0)r( f FSR /2 + f )∗ + r( f FSR /2)r( f )∗ = 2P0 J0 (Γ)J1 (Γ) . ν( f ) if (C.21) To calculate the expected dc gain |D(0)|, we apply L’Hôpital’s rule to the fraction in ¯ equation C.21. We can write r( f ) ≃ ( ∂r/ ∂ f )¯ f . The derivative is 0 ¯ ¯ ) ( 2πir ′0 t2a r b 2L ∂r ¯¯ ∂r ∂Φ ¯¯ = 2π = 2π = , × i c f FSR (1 − r a r b )2 ∂Φ ∂ω ¯0 ∂ f ¯0 | {z } (C.22) ≡ r ′0 ¯ where Φ = 2ωL/c is the round-trip phase of the cavity, and hence r ′0 = ∂r/ ∂Φ¯0 is the derivative of the reflectivity with respect to the phase. Therefore, D(0) = −2P0 J0 (Γ)J1 (Γ) 2π r Ω r ′0 f FSR . (C.23) We now want to derive the frequency dependence of D( f ) for f ≪ f FSR . In this limit, we have r( f FSR /2 + f ) ≃ r( f FSR /2), so the numerator in equation C.21 becomes approximately [ ] r( f FSR /2) − r(0) + r( f )∗ , (C.24) which has 0 as its lowest-order zero and f p as its lowest-order pole. This means that equation C.21 behaves as a single-pole filter with dc gain |D(0)| and pole f p : D( f ) ≃ D(0) 1 + i f / fp . (C.25) 141 In the limit of critical coupling ( r a = r b ≡ r ) and high finesse (1 − r 2 ≪ 1), we have r ′0 = r/(1 − r 2 ) = F/π = f FSR /2π f p and r Ω ≃ 1, so D(CC) (0) = −2P0 J0 (Γ)J1 (Γ)/ f p . (C.26) On the other hand, in the case of strong overcoupling ( r b = 1), one finds r ′0 = 2F/π, and hence D(OC) (0) = −4P0 J0 (Γ)J1 (Γ)/ f p . (C.27) C.3 Michelson locking Here we examine rf Michelson locking. Again the input light is phase modulated with an index Γ at an rf frequency Ω. The field incident on the Michelson’s symmetric port is p 2E 0 cos[ω0 t + Γ cos Ω t] [ ] E0 iΓ iΓ ≃ p eiω0 t 1 + eiΩ t + e−iΩ t + cc. 2 2 2 E in (t) = (C.28a) (C.28b) Upon transmission through the Michelson, the field at the antisymmetric port is { ( [ ) ] ω0 ℓ− (ω0 + Ω) ℓ− iΩ t iE 0 iω0 t iω0 ℓ+ /c iΓ i(ω0 +Ω) ℓ+ /c E as (t) = p e e sin sin + e e c 2 c 2 [ } ] (ω0 − Ω) ℓ− −iΩ t iΓ (C.29) + ei(ω0 −Ω) ℓ+ /c sin e + cc. 2 c We assume that the interferometer is operated close to a dark fringe; that is, the microscopic phase offset δϕ− = ω0 δ ℓ− /c ≃ sin(ω0 ℓ− /c) is small. However, the macroscopic phase offset (Schnupp asymmetry) is large, so that we have non-negligible sideband transmission to the antisymmetric port. We want to keep all terms only to linear order in δϕ− . The carrier term itself is linear in δϕ− , so we will keep only the dc sideband terms. To that end, we note that (to zeroth order in δϕ− ) we have ( ) ( ) ( ) ( ) ω0 ℓ− (ω0 ± Ω) ℓ− Ω ℓ− ω0 ℓ− Ω ℓ− = sin sin cos ± cos sin c c c c c ( ) Ω ℓ− ≃ sin c ≡ Φ− , (C.30a) (C.30b) (C.30c) 142 so [ ] iE 0 iω0 t iω0 ℓ+ /c iΓ iΓ iΩ ℓ+ /c iΩ t −iΩ ℓ+ /c −iΩ t E as (t) ≃ p e e δϕ− + Φ− e e − Φ− e e + cc 2 2 2 p [ ] = − 2E 0 sin(ω0 t + ω0 ℓ+ /c) δϕ− − ΓΦ− sin(Ω t + Ω ℓ+ /c) . (C.31a) (C.31b) The power sensed by the photodiode is Pas = 〈E 2as 〉ω0 [ ]2 = P0 δϕ− − ΓΦ− sin Ω t [ ] = P0 δϕ2− − 2ΓΦ− δϕ− sin Ω t + Γ2 Φ2− sin2 Ω t (C.32a) (C.32b) (C.32c) with P0 = E 20 . Therefore, demodulating at Ω/2π yields 〈Pas 〉Ω = P0 Γ sin(Ω ℓ− /c)δϕ− . (C.33) 143 D Some algebra for the differential arm length plant This section records some algebra for transforming the optomechanical plant for detuned resonant sideband extraction from the Buonanno and Chen formalism to a version that is more friendly to hands-on control room measurements and calibration (chapter 5). We start with the transfer function extracted by Ward from the Buonanno/Chen input/output relations: v [( ) ( ) ]u iβ 2iβ 2iβ u 2Pbs ω2 t e 1 − r e cos ϕ cos ζ − 1 + r e sin ϕ sin ζ δP s s s 0 t [ ] ∝ 2 2 4i β 2i β δh 1 + r s e − 2r s e cos 2ϕ + (K/2) sin 2ϕ ωa + ω2 (D.1) with β = − arctan ω/ωa ≡ arctan(− x) and K= 8Pbs ω0 8Pbs ω0 1 = . 2 2 2 2 2 2 2 ML ω (ωa + ω ) ML ωa ω (1 + ix)(1 − ix) Note √ eiβ = 1 − ix . 1 + ix (D.2) (D.3) The prefactor in equation D.1 can be rewritten as √ ts v u √ 2Pbs ω20 t s 2Pbs ω0 /ωa 1 − ix u t . = 1 + ix ω2a (1 + ix)(1 − ix) 1 + ix (D.4) 144 The numerator in equation D.1 can be rewritten as ( ) ( ) 1 − ix 1 − ix 1 − rs cos ϕ cos ζ − 1 + r s sin ϕ sin ζ 1 + ix 1 + ix 1 − ix = cos(ϕ + ζ) − r s cos(ϕ − ζ) 1 + ix ] 1 [ = (1 + ix) cos(ϕ + ζ) − r s (1 − ix) cos(ϕ − ζ) 1 + ix [ ] cos(ϕ + ζ) − r s cos(ϕ − ζ) cos(ϕ + ζ) + r s cos(ϕ − ζ) = 1 + ix × . 1 + ix cos(ϕ + ζ) − r s cos(ϕ − ζ) (D.5a) (D.5b) (D.5c) (D.5d) The denominator can be rewritten as ] ( )2 [ 1 − ix α 2 1 − ix sin 2ϕ 1 + rs − 2r s cos 2ϕ + 2 (D.6a) 1 + ix 1 + ix x (1 − ix)(1 + ix) [ ] 1 α 2 2 2 = (1 + ix) + r (1 − ix) − 2r (1 − ix)(1 + ix) cos 2 ϕ − 2r sin 2 ϕ (D.6b) s s 2 s (1 + ix)2 x {[ ][ ] ( ) } 1 2iϕ −2iϕ 2iϕ −2iϕ α = (1 + ix) − r e (1 − ix) (1 + ix) − r e (1 − ix) − ir e − e s s s (1 + ix)2 x2 (D.6c) { } [ ] 1 α (D.6d) = [(1 − p) + (1 + p)ix] (1 − p∗ ) + (1 + p∗ )ix − i(p − p∗ ) 2 2 (1 + ix) x ][ ] { [ } 1 1 + p∗ 1+ p ∗ α ∗ = 1 + ix × − i(p − p ) 2 (D.6e) (1 − p)(1 − p ) 1 + ix × 1− p 1 − p∗ (1 + ix)2 x ][ ] {[ } α (1 − p)(1 − p∗ ) 1 + p∗ i(p − p∗ ) 1+ p = 1 + ix × − (D.6f) 1 + ix × , 1− p 1 − p∗ (1 − p)(1 − p∗ ) x2 (1 + ix)2 where p = r s e2iϕ and α = 4Pbs ω0 /(ω4a ML2 ). Therefore, an equivalent representation of equation D.1 is √ t s 2Pbs ω0 cos(ϕ + ζ) − r s cos(ϕ − ζ) δP ∝ × δh ωa 1 − 2r s cos 2ϕ + r 2s     cos( ϕ + ζ ) + r cos( ϕ − ζ ) i f   s     1+ ×   f a cos(ϕ + ζ) − r s cos(ϕ − ζ) × [ . 2 2iϕ ] [ −2iϕ ]  f 1 + r e i f 1 + r e 2 α r sin 2 ϕ i f   s s s a     1+ × −   1+ × f a 1 − r s e2iϕ f a 1 − r s e−2iϕ 1 − 2r s cos 2ϕ + r 2s f 2 (D.7) We can tidy this up by defining the complex RSE pole p = fa × 1 − r s e2iϕ 1 + r s e2iϕ (D.8) , the square of the spring frequency ξ2 = f a2 × 2α r s sin 2ϕ 1 − 2r s cos 2ϕ + r 2s , (D.9) 145 which may be positive or negative, and the (real) RSE zero z = fa × cos(ϕ + ζ) − r s cos(ϕ − ζ) cos(ϕ + ζ) + r s cos(ϕ − ζ) (D.10) so that equation D.7 becomes √ / ] [ 1+if z t s 2Pbs ω0 cos(ϕ + ζ) − r s cos(ϕ − ζ) δP / )( / ) / ∝ × × ( . (D.11) δh ωa 1 + i f p 1 + i f p∗ − ξ2 f 2 1 − 2r s cos 2ϕ + r 2s 146 E In-situ test mass ringdown measurement Here we present an in-situ estimate of the coating and substrate thermal noise of an Advanced LIGO test mass, determined from the ringdown times of several of the test mass body modes. The loss angle ϕ = 1/Q of a particular test mass mode is related to the loss angle of the substrate and of the coating by ϕ= Us ϕs + Uc ϕc , Us + Uc (E.1) where Us is the portion of the mode’s strain energy stored in the substrate, and Uc is the portion in the coating. Because the volume of the coating is much smaller than the volume of the substrate, it is almost always the case that Us ≫ Uc , and in this situation we can say to very good approximation ϕ = ϕs + Uc ϕc . Us (E.2) If we know the loss angles and energy configurations of several modes, we can estimate the underlying loss angles ϕs and ϕc . Three modes were measured: a butterfly mode, the lowest-order drumhead mode, and the next-lowest-order drumhead mode (with a horizontal nodal line). Each mode was excited electrostatically using the high-frequency actuation path normally used to damp parametric instabilities;168 the ringdown was observed in the differential arm length readout. The Q factor of each mode was determined with least-squares fitting and the uncertainty was determined by examining the χ2 of the fit. The energy configurations were determined by finite element analysis: a simple Comsol model of a cylindrical piece of fused silica (with dimensions matching the test mass) was used to extract the bulk energy Us of the optic (in joules) 147 f (Hz) Q ∂Uc / ∂z ( −1 ) m Us Butterfly 6054 2.53(26) × 107 13.3 Drumhead 1 8158 2.13(45) × 107 13.9 9830 7 3.3 Drumhead 2 4.69(59) × 10 Table E.1: Data for determining the test mass coating and substrate loss from measured end Y optic ringdowns. The frequencies and Q factors are measured, and the energy ratios are determined via finite-element analysis. and the surface energy ∂Uc / ∂z of the optic face (in joules per meter). The energy in the coating is then Uc = ( ∂Uc / ∂z)d , where d = 6.2 μm is the thickness of the end test mass coating. The measured frequencies, modeled energy configurations, and measured Q factors are given in table E.1. To estimate ϕs and ϕc , we now turn to Bayesian analysis. As usual, our goal is to compute a posterior probability on ϕs and ϕc given our measurements {ϕ i }: ¯ ( ) ¯ ( ) p {ϕi }¯ϕs , ϕc p(ϕs , ϕc ) ¯ p ϕs , ϕc {ϕi } = . (E.3) Z Since in principle either ϕc or ϕs could range over several orders of magnitude, we choose the log-uniform prior p(ϕs , ϕc ) = 1 . ϕs ϕc (E.4) Our log-likelihood function is ( ) ∑ ϕ i − ϕ̂ i 2 ¯ ) ln L ∝ ln p {ϕ i }¯ϕs , ϕc ∝ − , 2σ2i i ( (E.5) where ϕ i is the measured loss angle of mode i , σ i is the uncertainty of the measurement, and ϕ̂ i = ϕs + ∂Uc / ∂z d ϕc . Us (E.6) Here we are assuming that both the coating and the substrate losses are structural. Our overall log-probability function for the Monte–Carlo simulation is then proportional to ln L + ln p(ϕs , ϕc ). The simulation yields the joint posterior shown in figure E.1. The 1D marginalizations of the posterior yield the following median values (with 1σ uncertainties): ϕs = 1.56(39) × 10−8 and ϕc = 2.8(8) × 10−4 . (E.7) 0 3. 0 5 1. 6. 0 4. 5 2 3. 4. 4 2. 0 6 1. φs × 108 3. 8 0. 1. 5 φc × 104 4. 5 6. 0 148 φc × 104 Figure E.1: Posterior pdf of Advanced LIGO test mass loss angles ϕs and ϕc , as determined from end Y ringdown data. Dashed lines in the 1D (marginalized) posteriors show 16th, 50th, and 84th percentiles of the posterior. The solid contours in the 2D (joint) posterior show 0.5σ, 1σ, and 1.5σ contours. With our knowledge of ϕs and ϕc , we can estimate the expected coating thermal noise level in Advanced LIGO. Implicit in our finite-element analysis is the assumption that the coating has the same material properties as the substrate. This is not correct (nor is it correct to speak of a single “loss angle” common to both bulk and shear strain),89 but nonetheless it is consistent with the theoretical analysis of Nakagawa et al.,169 who computed the Brownian noise of a test mass coated with a single lossy layer whose material properties match those of the substrate. In terms of displacement noise, their result is ] [ 2 (1 − 2σ) d ϕc . S xx ( f ) = ϕs + 1/2 (1 − σ) w π3/2 f wE π k B T(1 − σ2 ) (E.8) With E = 72 GPa, σ = 0.17, w = 62 mm, and T = 300 K, and with the loss angles 149 inferred in this appendix, this yields 1/2 S xx ( f ) [ −21 = 8.1(8) × 10 1/2 m/Hz ] ( ) 100 Hz 1/2 × f (E.9) for the end Y test mass. This is somewhat higher than—but consistent with—direct audio-band measurements by Gras et al.90 on end test mass coating witness sam[ ] ( / )1/2 ples, where the reported value is S xx ( f )1/2 = 7.1(3) × 10−21 m/Hz1/2 × 100 Hz f . 150 Bibliography 1. J. D. E. Creighton and W. G. Anderson. Gravitational-Wave Physics and Astronomy. Wiley-VCH, 2011 (cit. on pp. 1, 4). 2. A. Zee. Einstein Gravity in a Nutshell. Princeton University Press, 2013 (cit. on p. 1). 3. M. Bartelmann. The dark universe. Rev. Mod. Phys. 82 (1) (2010), pp. 331– 382 (cit. on p. 2). 4. F. Acernese, M. Alshourbagy, P. Amico, F. Antonucci, S. Aoudia, P. Astone, S. 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