Optical Resonators for Laser Frequency Stabilization and Tests of Lorentz Invariance

Transcription

1 Optical Resonators for Laser Frequency Stabilization and Tests of Lorentz Invariance Fred Baynes B.Sc Hons. This thesis is presented for the degree of Doctor of Philosophy at the University of Western Australia 23 rd September 2011 School of Physics The University of Western Australia Supervisors: Prof. Andre Luiten Prof. Michael Tobar

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3 Abstract This thesis is comprised of two parts: The first part details the implementation of two room temperature sapphire Fabry-Perot cavities as an optical frequency reference. The second part presents the first optical odd-parity test of Lorentz invariance, using an asymmetric optical resonator. We draw from the developments in the first part of the thesis to maximize the performance of the odd-parity test of Lorentz invariance. While sapphire presents material properties superior to the ULE and fused silica commonly used in Fabry-Perot resonators, the non-zero coefficient of thermal expansion of sapphire necessitates a high degree of temperature control. The optical cavities are housed in separate vacuum chambers with multi-stage temperature control reaching nanokelvin stability over a second. To frequency lock the 1064 nm laser to the stable Fabry-Perot cavities we use the Pound-Drever-Hall technique and employ Acousto-Optic Modulators (AOM) for fast frequency control. An evaluation of AOMs for this role is performed with no addition of noise above the 1 Hz/ Hz level. A systematic characterisation of noise sources in the optical frequency reference are performed and improvements through increased bandwidth and fast intensity control are undertaken. The fractional frequency stability of the sapphire Fabry-Perot resonators was limited by laser intensity noise at / Hz Odd-parity experiments are a new development in the field of tests of Lorentz invariance. Odd parity experiments offer significantly enhanced sensitivity to the odd-parity κ o+ and isotropic κ tr Lorentz violating parameters of the standard model extension, compared to the even-parity, Michelson-Morley type experiments recently undertaken. This experiment, the first optical odd-parity test of Lorentz invariance, uses an asymmetric ring resonator with a Brewster angled dielectric prism in one arm of the resonator. A high level of immunity to environmental fluctuation is provided by searching for violation of Lorentz invariance in the frequency difference of counter-propagating modes, a major advantage of this approach. Using the asymmetric ring resonator an experiment stationary in the laboratory marginally improved the constraint on the isotropic shift of the speed of light κ tr over the previous result. An orientation modulated experiment further improved the constraint, yielding κ tr =0.6 ± , a factor of 6 smaller constraint than reported before. i

4 Statement of Candidate Contribution The papers generated from the work contained in this thesis, and the author contribution to each, are outlined below: 1. Fred N. Baynes, Andre N. Luiten and Michael E. Tobar Testing Lorentz Invariance Using an Asymmetric Optical Resonator Physical Review D - Rapid Communications 84, (2011) The entirety of the experiment and data analysis was carried out by the author, under the supervision of Prof. Luiten and Prof. Tobar. This paper is included as Appendix A. 2. Anna Lurie, Fred N. Baynes, James Anstie, Phillip S. Light, Fetah Benabid and Andre N. Luiten. High-performance Iodine Fibre Frequency Standard Optics Letters 36, 4776 (2011) The author provided the stable reference laser locked to a Fabry-Perot cavity, generated and recorded the measurements of frequency stability. This paper is included as Appendix B. 3. Fred N. Baynes, Andre N. Luiten and Michael E. Tobar Oscillating Test of the Isotropic Shift of the Speed of Light Physical Review Letters, accepted for publication 5 th May 2012, manuscript LK The entirety of the experiment and data analysis was carried out by the author, under the supervision of Prof. Luiten and Prof. Tobar. The pre-print of this paper is included as Appendix C. Signed: Candidate: Signed: Coordinating Supervisor: ii

5 Acknowledgements Firstly I would like to thank Andre Luiten, whose passion, intuition and contagious enthusiasm for physics is unmatched. With an invariably correct solution to any problems encountered Andre has guided me through the process of a PhD with all the support and advice I could want. I would also like to thank Mike Tobar, whose idea is responsible for the commencement of this work and whose pragmatic attitude kept me motivated and on track. The post-docs in the group John McFerran, Clayton Locke, Jimmy Anstie and Phil Light have provided indispensable advice and equipment through the years and for this I thank them enormously. I owe special gratitude to Sam Dawkins and Paul Stanwix for the work that I have built on. The fellow students from my honours year Rhet, Paul, Dougie and Anna deserve mention for the mutual support we have provided each other over the years, as do the other students in the lab Gar-wing, Chris, Keal, Jerome and Weng. Everyone in the physics building deserves a mention from the workshop in the basement to the head of department for making it such a great place to work. For providing a life outside the lab I have to thank my good friends Campbell, Harry and James, it s always a good time when we get together. From the Lawler St house I should thank Tim, Thomas and Roberta for tolerating me and the good times we had. Everyone who I consider a friend (and that of course includes everyone in the acknowledgments, and more) has my sincere gratitude for making the life of a PhD student so much more than an experiment and a lab. The support of my family have been instrumental in my life and they have always been there to help me in anyway they can. They have encouraged me to do give my best and enjoy what I do, and this undertaking would not have been possible without them. Thankyou, Mum and Dad (and Isobel). Lastly I wish to thank Zoe, with all my love. Overflowing with positivity and affection she never fails to make me feel better. With patience and endless enthusiasm she has supported me through the years, and I am lucky to have her in my life. iii

6 To my Grandad who passed away during the writing of this thesis The original Fred Baynes with a love of Physics. iv

7 Table of Contents Abstract Statement of Candidate Contribution Acknowledgements i ii iii 1 Introduction Background Optical Atomic Clocks Developments in Fabry-Perot Optical Resonators Optical Resonator Experiments Tests of Lorentz Invariance Thesis Outline Sapphire Fabry-Perot Optical Cavities Odd-Parity Test of Lorentz Invariance Temperature Control of a Fabry-Perot Cavity Optical Resonators Sapphire Fabry-Perot Resonators Resonator Description Previous Limitations Temperature Control Heat Shields Microprocessor Temperature Controller Thermal Transfer Function Sapphire Cavity External Time Constant Heat Shield External Time Constant Heat Shield Internal Time Constant Temperature Stability Conclusions Frequency Control of a Laser Frequency Stabilization with an Optical Cavity v

8 TABLE OF CONTENTS 3.2 Control Loops Initial Optical Setup OFR 1 Frequency Lock - Piezo OFR 2 Frequency Lock - AOM Measuring Frequency Stability Acousto-Optic Modulators for Frequency Control Frequency Noise Floors AOM Frequency Stability Acoustic Phase Delay Induced Amplitude Modulation Conclusions Frequency Stability Noise Sources in Optical Resonators Vibration Bandwidth Limits Dual AOM setup Detection and Intensity Noise Fundamental Limits Brownian Motion Thermo-Elastic Damping Laser Shot Noise AOM for Fast Laser Intensity Control Use as a Frequency Reference Conclusions Tests of Lorentz Invariance Background The Standard Model Extension Choice of Reference Frame Odd-Parity Asymmetric Resonator Sensitivity to the SME Experimental Constraints on the SME parameters vi

9 TABLE OF CONTENTS Even-Parity Experiments Constraints on κ tr Proposed Experiment Stationary Odd-Parity Test of Lorentz Invariance Asymmetric Optical Resonators Counter-Propagating Modes Modulation Frequencies and the Short Data Set Approximation The Short Data Set Approximation Experimental Setup - Ring Resonator Experimental Setup - Optics and Frequency Locks Inherent Systematic Effects The Sagnac Effect Non-Reciprocal Effects Mode Coupling Data Analysis and Results Limitations of Sensitivity Conclusions Orientation Modulated Odd-Parity Test of Lorentz Invariance Orientation Modulation Rotation Stage Alignment Control Experimental Operation Demodulation and Results Future Odd-Parity Tests of Lorentz Invariance Conclusion Sapphire Fabry-Perot cavities Temperature Control Frequency Control Performance of the Frequency Reference vii

10 8.2 Odd-Parity Test of Lorentz Invariance Asymmetric Ring Resonator Counter-Propagating Modes Stationary Odd-Parity Test of Lorentz Invariance Orientationally Modulated Odd-Parity Test of Lorentz Invariance A Appendix 163 Paper: Testing Lorentz Invariance Using an Odd-Parity Asymmetric Optical Resonator B Appendix 169 Paper: High-performance Iodine Fiber Frequency Standard C Appendix 173 Paper: Oscillating Test of the Isotropic Shift of the Speed of Light viii

11 List of Figures 2.1 A simple Fabry-Perot optical resonator An example of reflected intensity as the laser frequency is swept over multiple FSR Picture and diagram of the sapphire Fabry-Perot Resonators Previous temperature stability of the sapphire Fabry-Perot resonators The large aluminium heat shield containing the sapphire cavity Thermal shielding and temperature control The AC Wheatstone bridge used for a temperature measurement The AC temperature control system The cavity external time constant τ ext with the heat flow paths through radiation R rad and conduction R cond Simulated thermal transfer function of a low-pass filter Sapphire cavity response to a change in temperature of the aluminium heat shield Thermal transfer function of the aluminium heat shield The aluminum cube Simulated transfer function Comparison of the old and new temperature control Square root allen variation of cavity temperature fluctuations Behavior of the reflection coefficient near resonance for an optimally coupled cavity The error signals after demodulation for fast and slow frequency modulation The absolute value of the derivative at resonance for the two error signal quadratures The frequency discriminator slope as a function of sideband power A simple control loop An example transfer function of a PI controller ix

12 LIST OF FIGURES 3.7 A simple control loop with noise The free-running fluctuations in the laser frequency Simplified mode-matching optics Reflected beam isolation The Initial frequency control setup The frequency locking scheme for OFR The transfer function of the piezo-electric transducer Acousto-optic Modulator Acousto-optic Modulator in the double pass configuration The locking electronics for OFR The transfer function of the frequency control system for OFR The laser frequency in the two arms showing the correction to the AOM is equivalent to a beat note Experiment to measure the optical beat note Frequency fluctuations from the correction voltage and an optical beat note Using the AOM correction signal to determine the frequency stability Measuring the input noise of the signal generators The FM input noise of two Agilent N181A signal generators The experimental setup to measure frequency fluctuations caused by the AOMs Frequency fluctuations imparted on the light by two double pass AOMs driven at constant frequency The propagation of acoustic waves in the AOM crystal Experiment to measure the acoustic phase delay of the AOM Acoustic phase shift as a function of AOM driving frequency Unwanted frequency noise caused by the acoustic phase delay in the AOM crystal Output intensity as a function of AOM driving frequency AM and FM conversion by the AOM x

13 LIST OF FIGURES 3.32 Increase in RIN when the AOM is actively controlling the laser frequency The in loop noise sources introduced by the AOM when used for active frequency control Dominant sources of noise in a laser frequency stabilized with an optical resonator Relative frequency Stability of the two Fabry-Perot Cavities Simple Interferometer to measure the motion of the Fabry-Perot cavity PSD of displacement of the input mirror of the Fabry-Perot cavities Frequency fluctuations from the motion of the Fabry-Perot cavities Susceptibility of the experiment to vibration Evaluation of the unity gain point Optical setup with two AOMs Frequency control setup for AOM Frequency correction sent to AOM Improved fast frequency stability of the dual AOM setup Noise floors from the Demodulations system and photodiodes Total noise floor of the frequency stabilization system AOM for fast intensity control VVA and AOM transfer function Transfer function of the Sallen-Key bandpass filter In-loop and out of loop reduction of laser RIN by a VVA and AOM using a band-pass filter with Q Transmission intensity control loop SRAV of transmitted power with control loop locked and unlocked Measurement of frequency stability of HC-PCF iodine frequency reference SRAV of the frequency stability of the iodine HC-CPF reference Fractional frequency stability of the two Fabry-Perot optical cavities Perspective and plan of the Michelson Morley experiment xi

14 LIST OF FIGURES 5.2 Sun-centered Celestial Equatorial reference frame (SCCEF) used in the SME. The SCCEF has the Z axis pointing towards the celestial north pole, the X axis points from the sun towards the earth at the moment of the autumnal equinox, and the Y axis is chosen in accordance with the right hand rule A generic asymmetric traveling wave ring resonator Improvement of the constraints on ( κ e ) XY Direct Measurements of κ tr The asymmetric optical ring resonator The orientation of the fields for vertical ring resonator Counter-propagating modes in an asymmetric ring resonator The asymmetric optical ring resonator Experimental setup for stationary test of Lorentz invariance PSD of fractional frequency fluctuations in a single propagating mode of the asymmetric ring resonator Diagram of the orientation of the area vector of the asymmetric ring resonator Mode coupling induced splitting of a resonance Processed time series data Power spectral density Values of κ tr determined from ordinary least squares regression DC offset in a PDH error signal caused by RAM Long term frequency drift and mode matching variations Orientation modulation of the experiment SRAV of the stationary test of LI Photograph of the orientation modulated test of Lorentz invariance Vibration of the experiment during rotation Diagram of alignment control Alignment control error locked and unlocked Recorded motion of the experiment xii

15 LIST OF FIGURES 7.8 Demodulated time series data Values of κ tr determined from ordinary least squares regression for the orientation modulated experiment Power spectral density of the residuals from the ordinary least squares Bow-tie resonator for a rotating test of Lorentz invariance Comparison of Results xiii

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17 The power to question is the basis of all human progress Indira Gandhi ( ) 1 Introduction 1.1 Background Fundamental tests of physics question our understanding of nature at the most profound level. Physics, the framework describing the mechanics of the universe, must conform to experimental reality. For physics to advance existing theories need to be tested at the limits of technology and the predictions of new theories require experimental verification. Particle accelerators such as the large hadron collider at CERN use extremely high energies to generate interactions that may reveal the limitations of current theories. Alternatively, at low energies precision measurements seek to perform experiments with increasing sensitivity so that minute deviations from the expected result, or a new effect, can be observed. At the forefront of precision measurement is the performance of experiments measuring and controlling the frequency of light. Experiments based on optical frequency metrology have provided some of the most precise tests of our understanding of the universe. Advances made during such research also drives progress in our technology-based society as developments made during experiments of fundamental physics are implemented on a global scale. Optical resonators are used exclusively, or comprise a key component, in many of these experiments and represent an important area of research. Optical resonators are at the most elementary level, a set of high reflectivity mirrors designed to cause light to propagate in a closed path. These mirrors can be arranged linearly; where the light is constantly reflected back and forth between two mirrors facing each other, or in a ring; where the light propagates back onto itself through some closed loop. Resonance occurs when an external light source, with 1

18 CHAPTER 1. INTRODUCTION the correct wavelength, leaks into the resonator and builds up an intense circulating field through constructive interference. These optical resonances can be detected and used to define a precise frequency or sense miniscule variations in the resonant electro-magnetic fields Optical Atomic Clocks The modern generation of optical atomic clocks is the most precise measurement tool available to science, with uncertainty at the level [Rosenband 08]. The electromagnetic transitions of atoms, isolated from the environment, provide extremely stable and pure resonances that can be interrogated to measure the frequency of electromagnetic radiation. From the definition of the second as the transition between the two hyperfine levels of the ground state of the caesium 133 atom [CGPM 67], atomic clocks provide the primary standard for the measurement of time and international time keeping networks. Optical atomic clocks have been used to test general relativity [Chou 10] and search for variations in the fundamental constants [Bize 03, Fortier 07, Blatt 08]. While the performance of optical atomic clocks is outstanding, they have not reached the potential implied by quantum mechanics. In neutral atom clocks the stability is limited by the performance of the clock laser to several orders of magnitude above the fundamental limit of quantum projection noise [Jiang 11]. The clock laser is generally a laser frequency locked to a high finesse ultra-stable optical resonator, taking advantage of the optical resonator s excellent short term stability. A stable clock laser is required because optical atomic clocks have time-separated preparation, detection and interrogation phases and require a clock laser to maintain the oscillator phase between interrogation phases. The clock laser must be stable enough to vary by less than 1 radian between interrogations of the atomic transition, and on timescales shorter than the measurement cycle the stability of the optical atomic clock is dictated by the stability of the clock laser. Furthermore, the Dick effect [Quessada 03] which arises from the periodic probing of the atomic resonance, transfers the noise of the clock laser into the output of the optical atomic clock, 2

19 1.1. BACKGROUND limiting the stability that can be achieved. Advances in the stability of clock lasers locked to optical resonators can directly enhance the performance of optical atomic clocks Developments in Fabry-Perot Optical Resonators The typical optical resonator used to frequency stabilize lasers for atomic clocks are Fabry-Perot cavities, which consist of two high reflectivity mirrors on either end of a linear spacer. The resonant frequency is defined by the distance between the two mirrors and fluctuations in the dimensions of the resonator reduce the frequency stability of the laser locked to the optical cavity. The spacers are generally made of Ultra Low Expansion (ULE) glass, which has a near zero coefficient of thermal expansion above room temperature. By suitable temperature control of the Fabry-Perot cavity the length fluctuations caused by temperature variations can be minimized. While this a major advantage of ULE cavities, the relatively low Young s modulus of of ULE make the cavities susceptible to vibration. By judicious design of both the cavity geometry and mounting it is possible to substantially reduce the sensitivity to vibration [Notcutt 05, Chen 06a]. Finite element analysis has refined the technique and produced a variety of cavity shapes and mounting systems [Nazarova 06, Webster 07, Zhao 09, Millo 09, Webster 11] with reported sensitivity to vibration as low as ms 2. Fundamental noise sources in optical resonators also contribute to the ultimate stability of the devices. An optical cavity at finite temperature will suffer from thermodynamic fluctuations such as Brownian motion that continuously varies the dimensions of the resonator. The ULE glass used in the construction of the cavities has a high mechanical loss, and this increases in the contribution of Brownian noise to stability of the cavity [Numata 04]. The limiting factor in the stability of Fabry- Perot resonators has been confirmed as Brownian motion in the cavity components [Numata 04, Notcutt 06, Ludlow 07]. The substitution of ULE mirrors with fused silica, which has a lower mechanical loss, has recently been introduced to enhance the performance of Fabry-Perot cavities [Millo 09, Legero 10, Jiang 11]. In addition 3

20 CHAPTER 1. INTRODUCTION to Brownian motion, there can also be a contribution to thermal noise from thermoelastic damping which will arise from materials in the resonator with a non-zero coefficient of thermal expansion. This noise source is yet to be seen directly in Fabry-Perot cavities but it is likely to play a role in the noise sources for gravity wave detectors [Braginsky 99, Braginsky 03, Harry 06] Optical Resonator Experiments As a highly sensitive tool for scientific advancement optical resonators are directly used for a variety of fundamental tests. Gravity wave observatories attempt to detect the minute ripples in space-time predicted by Einstein s general relativity. The most sensitive gravity wave detectors, such as LIGO, VIRGO and GEO 600, use massive laser interferometers to detect the motion of suspended free masses as a gravitational wave passes. Optical resonant cavities are used in the interferometer arms to improve the sensitivity of the detectors [Strain 91]. A Fabry-Perot cavity has also been employed in an experiment designed to measure the birefringence of the vacuum caused by electron-positron vacuum fluctuations [Valle 10]. The use of optical resonators most pertinent to this work is in the search for violations of Lorentz invariance as described below Tests of Lorentz Invariance In accordance with special relativity and Lorentz invariance, the speed of light is a fundamental property of the universe and is assumed to be constant in all reference frames and at all velocities. Attempts to reconcile the standard model of particle physics with general relativity have produced theories predicting a breakdown of Lorentz invariance at some level [Kostelecký 89, Kostelecký 91, Alfaro 02, Bjorken 03]. The standard model extension [Colladay 98, Colladay 97] serves as the theoretical frame work in which modern tests of Lorentz invariance are analyzed: it parameterizes all possible violations of Lorentz invariance by known fields. Increasingly stringent experimental constraints on possible violations of Lorentz invariance can disprove speculative unification theories, and any detected violation of Lorentz 4

21 1.2. THESIS OUTLINE invariance would have tremendous implications for the direction and development of physics. The history of experimental tests of Lorentz invariance is rich, beginning with the famous work of Michelson and Morley [Michelson 87] in With the advent of modern frequency standards the limits on possible violations of Lorentz invariance have improved by six orders of magnitude in the past 50 years. Modern tests of Lorentz invariance make use of optical resonant cavities as the resonant frequency of an optical cavity is directly related to the speed of light. Modern tests of Lorentz invariance have reached extraordinary precision using Fabry-Perot optical resonators [Eisele 09, Herrmann 09]. These modern Michelson-Morley type, even-parity experiments consist of two orthogonally mounted, standing wave optical Fabry-Perot resonators. While sensitive to certain types of violations of Loretntz invariance, the condition of resonance and symmetries of these experiments lead to suppressed sensitivity to some of the Lorentz violating parameters contained in the standard model extension [Tobar 05]. 1.2 Thesis Outline This thesis is comprised of two parts: The first part - chapters 2, 3 and 4 detail work on a room temperature sapphire Fabry-Perot optical cavity. The second part - chapters 5, 6 and 7 are the results of the first odd-parity test of Lorentz invariance using an asymmetric ring resonator. We exploit the technologies developed in the first part to optimize the control systems necessary for the second part Sapphire Fabry-Perot Optical Cavities The optical resonators used in the first series of experiments is a pair of 15 cm linear Fabry-Perot cavities with crystalline sapphire spacers. Sapphire has superior material properties to ULE and fused silica. It has a higher Young s modulus and lower mechanical losses than both and does not exhibit material creep. The disadvantage is the non-zero coefficient of thermal expansion at room temperature of sapphire and the cavities require extensive temperature control. The non-zero coefficient 5

22 CHAPTER 1. INTRODUCTION of thermal expansion does however mean the cavities should exhibit thermo-elastic damping noise and one of the motivations of the experiments were attempts to reach the fundamental stability limits of the sapphire Fabry-Perot cavities and measure the existence of thermo-elastic damping noise. Another primary motivation was the requirement of a stable frequency reference for other experiments being performed in the laboratory together with the development of high performance frequency locking and noise characterisation techniques. A 1064 nm laser was locked to the sapphire Fabry-Perot cavities using the Pound-Drever-Hall frequency locking technique [Drever 83, Salomon 88] to act as the precision reference for noise determination and frequency comparison. Chapter 2 details the temperature control system required to stabilize the temperature of the sapphire cavities. A microprocessor temperature controller based on an AC Wheatstone bridge was used for multi-layer temperature stabilization and the cavity housing was designed to maximize the thermal isolation of the resonators. The thermal characteristics of the experiments were determined with thermal time constants approaching two days. The measured temperature of the sapphire Fabry-Perot cavities was 3 nk/s corresponding to a fractional frequency stability of In chapter 3 the frequency locking technique is presented, we use acousto-optic modulators in a configuration enabling independent frequency locks for the two cavities and negating the need for an optical beat note. Extensive measurements were performed to evaluate the performance of the acousto-optic modulators for frequency control and no additional noise above 1 Hz/ Hz was identified. Chapter 4 is a systematic evaluation of the fundamental and technical noise sources in the frequency stabilization scheme and Fabry-Perot cavities. Mechanical vibration of the cavities was identified as one limit to the frequency stability. To improve the noise suppression of the frequency control loop an acousto-optic modulator was used to for fast control, increasing the bandwidth from 7 khz to 100 khz. The limit of the PDH frequency stabilization was found to be laser intensity noise at the modulation frequency, limiting the fractional frequency stability to the level of / Hz at 1 khz, within an oder of magnitude of the fundamental limits. 6

23 1.2. THESIS OUTLINE To improve this result the use of a second-order control loop was investigated. The frequency stabilized laser locked to the sapphire cavities was used as a frequency reference to measure the stability of an Iodine reference based on hollow core crystal photonic optical fibres Odd-Parity Test of Lorentz Invariance The apparatus and precision measurement techniques developed in the first part were applied to the first odd-parity optical test of Lorentz invariance. Recent developments in the methods of testing Lorentz invariance have shown the improved sensitivity of odd-parity experiments to certain types of Lorentz invariance compared to even-parity experiments [Tobar 05]. More specifically, within the framework of the standard model extension, even-parity experiments have direct sensitivity to the even-parity κ e Lorentz violating parameter and have suppressed sensitivity to the odd-parity κ o+ parameter and isotropic, scalar κ tr. The level of sensitivity reduction is the relative velocity of the earth, β Even parity experiments have first order β suppressed sensitivity to the odd-parity κ o+ parameter and second order β 2 suppression of sensitivity to the isotropic, scalar parameter κ tr. Thus odd-parity and isotropic violations are severely under-constrained compared to the even-parity violations of Lorentz invariance. Odd-parity experiments offer direct sensitivity to the odd-parity κ o+ and only first order suppression of the scalar κ tr. Therefore odd-parity experiments have a sensitivity 10 4 greater to odd-parity and isotropic violations of Lorentz invariance than even-parity experiments and the capacity to greatly improve the experimental constraints on these parameters. The odd-parity experiment used for these experiments is a traveling wave asymmetric optical ring resonator, with the asymmetry provided by the presence of a dielectric material in one arm of the resonator. The use of counter-propagating modes in the asymmetric ring resonator gives exceptional immunity to environmental fluctuations and enabled a new constraint to be placed on the isotropic standard model extension parameter κ tr. Chapter 5 is an overview of historical and modern tests of Lorentz invariance 7

24 CHAPTER 1. INTRODUCTION and an introduction of the standard model extension, a theoretical frame work for tests of Lorentz invariance. In chapter 6 the asymmetric ring resonator and counter-propagating modes are introduced. The experimental details and results of a stationary test of Lorentz invariance, relying on the rotation of the earth for modulation, are reported. Inherent systematic effects in the asymmetric ring resonator are determined to be several orders of magnitude below the expected sensitivity. The benefits of the counter-propagating modes enabled a relatively simple experiment to marginally improve on the previous constraint with κ tr =3.4 ± after 45 days of measurement. In chapter 7 we use orientational modulation of the asymmetric ring resonator to improve the results of the previous experiment. By reversing the orientation of the experiment every 617 seconds the signal for a violation of Lorentz invariance occurs at a much higher frequency where the experiment has improved frequency stability. Active intensity of alignment control suppress the systematic errors associated with the motion of the experiment. From over 6000 reversals the experiment over 66 days the constraint on the isotropic shift of the speed of light was improved by a factor of 6 to κ tr =0.3 ±

25 Change and decay in all around I see; O thou who changest not, abide with me. Henry Lyte ( ) 2 Temperature Control of a Fabry-Perot Cavity Chapter Overview Most frequency stabilized lasers have been based on stabilization of the laser frequency to a mode of a Fabry-Perot cavity. To achieve the best possible stability requires the optical resonator to be dimensionally stable. The sapphire Fabry-Perot cavities used in these experiments have a non-zero coefficient of thermal expansion at room temperature and the efforts to minimize the temperature fluctuations are presented here. In this chapter we first present an analysis of optical resonance and the relationship to the physical parameters of the cavity. The details of the temperature control system are described, we use an AC Wheatsone bridge to reduce systematic errors in the system. The experiment was designed to maximize the thermal isolation of the sapphire cavities and the thermal transfer function of the system was determined to measure the success of our approach. The results are compared to a simulation of heat flow and provides an insight into the application of an active temperature control setup. Finally the results of the temperature stabilization and corresponding frequency stability are reported. 2.1 Optical Resonators By way of introduction we consider the behavior of a light field in a Fabry-Perot optical resonator. A simple Fabry-Perot cavity consists of two mirrors, with ampli- 9

26 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY tude reflection r 1 and r 2 and amplitude transmission t 1 and t 2, separated by some distance L, see figure 2.1. In this example we will use neglect losses in the mirrors, so ri 2 +t 2 i = 1. Internal losses in the Fabry-Perot cavity over a round trip are introduced by the parameter a, where a lossless resonator has a =1. Figure 2.1: A simple Fabry-Perot optical resonator We will follow a method similar to [Riehle 04] by considering the amplitude of the electromagnetic wave reflected by the resonator. The incident beam has complex amplitude Ẽinc and angular frequency ω. The reflected beam is: Ẽ ref = Ẽincr 1 Ẽinct 2 1ar 2 e iλ Ẽinct 2 1a 2 r2r 2 1 e i2λ Ẽinct 2 1a 3 r2r 3 1e 2 3Λ +... = Ẽinc[r 1 at 2 1r 2 e iλ (1 + ar 1 r 2 e iλ +(ar 1 r 2 e iλ ) ] ] [r 1 at2 1r 2 e iλ = Ẽinc = Ẽinc 1 ar 1 r 2 e [ ] iλ r1 ar 2 e iλ 1 ar 1 r 2 e iλ where Λ is the phase shift associated with a round trip in the resonator. reflection coefficient is: 2.1 The F (ω) =Ẽref = r 1 ar 2 e iλ Ẽ inc 1 ar 1 r 2 e iλ Similar equations can be derived for the circulating and transmitted light: 2.2 Ẽ circ Ẽ inc = Ẽ trans Ẽ inc = t 1 1 ar 1 r 2 e iλ 2.3 at 1 t 2 1 ar 1 r 2 e iλ 2.4 For an incident field of power I inc the power of the reflected field is I ref = Ẽref 2 10

27 2.1. OPTICAL RESONATORS and is given by I ref = I inc [1 C where the contrast C of the resonator is: ] 1 1+ ( ) 2F 2 π sin 2 ( Λ) ( ) 2 r1 ar 2 C =1 1 ar 1 r In the case of an optimally impedance matched resonator the amplitude reflectivity r 1 of the input mirror matches the additional losses ar 2. For a perfectly impedance matched resonator, from equation 2.2, the amplitude of the reflected beam is zero and from equation 2.6 the contrast is 1. The finesse F is: F = π ar 1 r 2 1 ar 1 r A cavity is in resonance with the incoming beam when the round trip phase shift Λ is an integer multiple of 2π. For a Fabry-Perot resonator the round trip phase shift is: Λ= 2πν 2L c 2.8 where ν is the frequency of the light and L is the length of the resonator and we have ignored phase shifts on reflection from the mirrors. Therefore the frequency spacing between resonant modes (the Free Spectral Range - FSR) of the resonator is: ν FSR = c 2L 2.9 As the laser frequency is swept through the resonance of the cavity the reflected intensity I ref has the form of a Lorentzian curve and an example curve is illustrated in figure 2.2. The cavity used in this example has a finesse F = 100, no internal losses and r 1 = r 2 so the resonator is optimally impedance matched and C =1 The finesse of a resonator gives a measure of the ability of the resonator to discriminate frequency and can be expressed as the ratio of the frequency between resonances ν FSR divided the full width half maximum of the cavity resonances ν cav : F = ν FSR ν cav

28 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY Figure 2.2: An example of reflected intensity as the laser frequency is swept over multiple FSR. The cavity has a finesse of 100 and is optimally impedance matched In this experiment to frequency stabilize a laser with a Fabry-Perot cavity the frequency of the laser is locked to the resonant frequency of the cavity by the Pound- Drever-Hall technique [Drever 83] (see Section 3.1) and this forms the basis of the optical resonator frequency reference. The reflected beam is used to perform the Pound-Drever-Hall technique and it is desirable to have a high contrast and high finesse resonator. Furthermore, from equation 2.9 a laser that is locked to the resonant frequency can only be as stable in frequency as the cavity is stable in length, since fluctuations in cavity length correspond directly to changes in the resonant frequency: ΔL L = Δf f 2.11 Hence the resonant frequency of the cavity is dependent on the distance between the two mirrors. Thus, a good frequency reference requires a dimensionally stable Fabry- Perot cavity. The next section will outline attempts to stabilize the dimensions of a Fabry-Perot cavity through temperature control. 12

29 2.2. SAPPHIRE FABRY-PEROT RESONATORS 2.2 Sapphire Fabry-Perot Resonators The Fabry-Perot cavities that forms the basis of these experiments were used previously for the construction of a room temperature sapphire optical frequency reference. The initial design, construction and determination of the cavity parameters is documented in [Dawkins 07] and we will use many of the same values. Sapphire has a non-zero coefficient of thermal expansion and the temperature stability of the sapphire Fabry-Perot resonators was found to be a major limiting factor in the development of the experiment Resonator Description Two Fabry-Perot optical cavities were housed in separate temperature controlled vacuum environments using fused silica and sapphire mirrors attached to sapphire spacers, see figure 2.3. The sapphire spacers are 150 mm long and 50 mm in diameter with a 6 mm axial bore connected to a pump-out hole so the light circulates in a vacuum. The 150 mm length was chosen to give a nominal FSR of 1 GHz and the diameter allows the clamping of either 25 mm or 50 mm mirrors. The two cavities are referred to as OFR 1 and OFR 2 and both have low loss dielectric multi-layer coated mirrors, sourced from CSIRO [Blair 97]. The mirrors on OFR 1 are on a fused silica substrate, one having a radius of curvature of 500 mm and the other mirror is flat. Both have a diameter of 40 mm and a thickness of 10 mm. The mirrors on OFR 2 have a sapphire substrate with one mirror having a radius of curvature of 222 mm and the other is flat. The sapphire mirrors are 25.4 mm in diameter with a thickness of about 6 mm. The mirrors are clamped to the spacers with custom clamps designed to ensure that peak environmental accelerations are not able to dislodge the mirror [Dawkins 07]. The parameters of the cavities have been measured by S. Dawkins and are summarized in table 3.1. It should be noted that the lower finesse cavity OFR 1 is better coupled (has a higher contrast) than the lower finesse cavity. This has led to both cavities having approximately the same frequency discrimination so the 13

30 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY (a) (b) Figure 2.3: Picture (a) and diagram (b) of the sapphire Fabry-Perot resonators cavities delivered about the same level of performance. The relatively low finesse by contemporary standards is a consequence of the mirrors, which have nominal reflectivity of 98.95% and 99.99% on OFR 1 and OFR 2 respectively. Table 2.1: The cavity parameters from [Dawkins 07], confirmed experimentally OFR 1 OFR 2 FSR (ν FSR ) 1.02 GHz GHz Bandwidth (ν cav ) 840 khz 177 khz Finesse (F) Contrast (C) Previous Limitations The sapphire spacers and the sapphire and fused silica mirrors all have non-zero coefficient of thermal expansion so the temperature stability of the cavity is highly important. Environmental fluctuations cause the length of the cavity to change through thermal expansion: ΔL = αlδt 2.12 where ΔL is the change in length, α is the coefficient of thermal expansion, L is the length and ΔT is the change in temperature. The dominant source of length changes are from the sapphire spacer which has α sapph = K 1 [Nikogosian 97] 14

31 2.3. TEMPERATURE CONTROL at room temperature. This leads to a temperature dependent frequency shifts of Δf = α sapph f Hz/K for 1064 nm ( Hz) light. This level of temperature sensitivity requires strict temperature control of the environment and the previous attempts were found to be inadequate. The two cavities were mounted in the vacuum with copper clamps on stainless steel posts, with more stainless steel posts connecting the two base plates to the outer vacuum chamber. There were two heat shields made of thin (< 1 mm thick) copper sheets that were temperature controlled through a DC measurement, microprocessor based temperature control system. This level of thermal shielding and temperature control was insufficient for the sapphire cavities, with temperature drifts of 0.25 μk/s and corresponding frequency drifts of 400Hz/s, highly correlated with daily laboratory temperature fluctuations [Dawkins 07], see figure 2.4 below. Figure 2.4: Previous temperature stability of the sapphire Fabry-Perot resonators. 2.3 Temperature Control To reduce the frequency drift of the sapphire Fabry-Perot resonators a series of major revisions were made to the experiment. The thermal shielding and temperature controller have been improved to increase the thermal stability of the experiment. 15

32 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY Heat Shields The sapphire cavities are now enclosed by thick and heavy aluminium heat shields, see figure 2.5. Aluminium was chosen for its high thermal diffusivity, κ Al = m 2 /s. Thermal diffusivity is equal to k/(ρc p ) where k is the conductivity of the material, ρ is the density and c p is the specific heat. It is essentially a measure of how quickly a material responds to temperature changes - a material with high thermal diffusivity responds faster and reaches equilibrium quickly with the stable temperature control point. The shape of the aluminum heat shields was intended to increase their thermal filtering ability, to shield the sapphire cavities from residual temperature fluctuations. They have a large thermal capacity and minimal surface area to volume ratio. The heat shields are short cylinders, 15 cm in diameter and with 2 cm thick walls and weigh about 5.8 kg. The stainless steel posts of the previous setup were replaced with teflon (polytetrafluoroethylene) posts, since teflon has a thermal conductivity 64 times smaller than stainless steel (0.25 vs 16 W/m K). The two copper heat shields were left in situ to provide additional passive thermal filtering but the temperature control system on the inner shield was moved to the aluminium heat shield. Both control systems were improved and commercial temperature controllers were added to the two baseplates (Thorlabs TED200C and TC200 on the inner and outer baseplates respectively). There are now four independent temperature control points, see figure 2.6. Figure 2.5: The large aluminium heat shield containing the sapphire cavity. 16

33 2.3. TEMPERATURE CONTROL Figure 2.6: Thermal shielding and temperature control. (1) Sapphire cavity, (2) Temperature controlled Aluminium heat shield, (3) Teflon posts, (4) Passive inner copper heat shield, (5) Temperature controlled outer heat shield, (6) Temperature controlled inner baseplate, (7) Temperature controlled outer baseplate, (8) Vacuum 10 5 Torr throughout the experiment. Red triangles indicate temperature control and blue squares indicate the position of thermistors Microprocessor Temperature Controller A single board microprocessor (Micro-PAC 180) with integrated analog ADC and DAC was used to implement an AC Wheatstone bridge and temperature controller, see figure 2.8. The sensors in the vacuum chamber for the temperature control systems are 10 kω thermistors that form part of a Wheatstone bridge, a highly sensitive way of converting the thermistor resistance to a voltage used as the input for the temperature control system. A voltage is passed through the bridge and the when the bridge is balanced (R 2 /R 1 = R th (T )/R 3, R th (T ) is a thermistor where the resistance R th is a function of temperature T) zero voltage will be measured across the bridge, see figure 2.7. We used the Digital to Analog Converter (DAC) on the microprocessor to drive the bridge with a 5 V square wave at 1.3 Hz and we used 17

34 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY Figure 2.7: The AC Wheatstone bridge used for a temperature measurement, R th is a thermistor in the vacuum system. a 2.5 V reference (LM336) to place a fixed voltage on the other end of the bridge. The Wheatstone bridge voltage is then amplified by a differential instrumentation amp (AD524AD) by a factor of 1000, added to the 2.5 V reference and input to the temperature control algorithm by the microprocessor s 12 bit Analog to Digital Converter (ADC). The conversion of resistance to voltage by a Wheatstone bridge is given by: V Wh = ( Rth (T ) R ) 2 V 0 R 3 + R th R 1 + R where V 0 is the driving voltage. In the Wheatstone bridge R 1 is a 10 kω precision resistor, R 2 is the set point at 8 kω made from 2 precision resistors in parallel and R 3 is a precision resistor with an extremely low temperature coefficient (0.05 ppm/ C). The 5V analog to digital input of microprocessor was 12 bits. Combining this with the equation 2.13 thermistor temperature response and amplifier gain we calculate the digitization noise error on the input to the microprocessor is ±25 μk. However, the internal representation of the temperature is averaged over 6 cycles giving the temperature controller 11 μk resolution. The microprocessor temperature control algorithm uses the infinite impulse response (IIR) method to determine the required voltage output. The immediate voltage output to the heater y n is calculated from the previous voltage outputs y n j 18

35 2.3. TEMPERATURE CONTROL and temperature inputs x n k : y n = N M a k x n k + b j y n j k=0 j= where the coefficients a k and b j are calculated to give the control algorithm the desired transfer function. The coefficients and gain where chosen to form a simple PI control loop (see section 3.2) with a break point (a change of slope in the transfer function) at Hz to match the measured transfer function of the heat shield and electronics. The temperature control algorithm controls the output of the DAC whicj is used to drive the Wheatstone bridge and it switches between 0 and 5 V at 1.3 Hz with a 2.5 V voltage reference providing a virtual ground. A measurement of the bridge voltage is taken with the effective bridge driving voltage at +2.5 V then -2.5 V, and the difference between them is calculated and averaged. By making an AC measurement this way any stray voltages such the thermo-electric effect in the leads to the thermistor will be subtracted away and only the true bridge voltage will remain. See figure 2.8 for an overview of the temperature control system. The outputs of the temperature controllers are passed through 1 Amp noninverting amplifiers to give the current necessary to deliver sufficient heat inside the vacuum chamber. The heaters on the outer copper shields are 20 m of resistance wire (Minalpha) coiled around the shield and varnished to the surface with the control thermistor. The heater on the aluminium heat shield is a single ceramic plate resistor firmly bolted with two thermistors (one used by the control loop and one for an outof-loop measurement) by a small copper plate to the surface of the heat shield. The thermistor and heater are as close as possible to each other and a small flat piece of indium metal was squashed between the heat/thermistor and the surface of the heat shield to ensure good thermal contact. To monitor the performance of the temperature control there is a 4-wire thermistor attached to the sapphire cavity and 2-wire thermistors on the inner aluminium shield, passive copper shield, controlled copper shield and the two base plates. The resistances of four of these thermistors were measured by digital multimeters (Agilent 34401A) and recorded every 4 seconds through a GPIB data acquisition system. The measurements taken 19

36 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY Figure 2.8: The AC temperature control system. The microprocessor supplies a 1.3 Hz 5 V square wave (1) to the two Wheatstone bridges (2) incorporating the 10 kω thermistors on the outer and inner heat shields (4). The 2.5 V reference (3) acts as a virtual ground and offset for the instrumentation amplifiers (5) whose output is fed into the microprocessor run temperature control algorithm (6). The output of the DAC from the temperature control algorithm is sent to the 1 A amplifiers (7) and finally to the heaters in the vacuum chamber (8). by these instruments allows us to form a model of the temperature control system. 2.4 Thermal Transfer Function To reduce temperature fluctuations of the sapphire Fabry-Perot cavities they have been thermally isolated and surrounded by temperature controlled heat shields. A measure of the thermal isolation of the sapphire Fabry-Perot resonators is the 20

37 2.4. THERMAL TRANSFER FUNCTION external time constant - how long it takes to reach thermal equilibrium with the external environment. A large external time constant filters residual temperature variations and reduces temperature changes in the cavities. The temperature control of the heat shields operate by maintaining a constant temperature at a specific point, and the time it takes for the heat shields to reach internal equilibrium with this point is the internal time constant. For active temperature control, such as that on the heat shields, it is desirable to have a large ratio between the external time constant and the internal time constant. In this situation the heat shields are thermally isolated from the environment and well thermally coupled to point with stable temperature. In other words, temperature gradients will be minimized and a uniform, stable temperature will be presented to the sapphire Fabry-Perot resonators Sapphire Cavity External Time Constant The sapphire resonators are housed in vacuum and heat can flow by radiation and conduction through the supports, see figure 2.9. The thermal resistance of the heat flow paths contribute to the external time constant of the sapphire cavities. Figure 2.9: The cavity external time constant τ ext with the heat flow paths through radiation R rad and conduction R cond. The thermal resistance from the heat conducted through the teflon support posts is given by [Holman 81]: R cond = ΔT Q cond = L k tef A 2.15 where L is the conductive length, k tef is the thermal conductivity of teflon and A is the cross-sectional area. The two teflon supports have cross-sectional area of 3 cm 2 21

38 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY and are 3 cm high which gives a total thermal resistance of 200 K/W. Radiation is the other possible route of heat transfer, with the radiative thermal resistance given by: R rad = ΔT Q rad = ΔT σεa(t 4 (T ΔT ) 1 4 4σεAT wherewehaveexpandedaboutδt to linearize the equation. Given the area of the cylinder A, the Stefan-Boltzmann constant σ and the emissivity of sapphire, (ε = 0.47, [Wittenberg 65]) and the temperature, T, the radiative thermal resistance of the cavity is 50 K/W. Combining these two results in parallel the total thermal resistance to the external environment is 40 K/W. To calculate the external time constant we use the relatively simple formula [Dratler 74]: τ ext = R ext C cav f ext = πτ ext where C is the total thermal capacity of the cavity, calculated by using the specific heat c sapph = 782 J/kgK [Nikogosian 97] to be 940 J/K. This gives an external time constant of τ ext seconds. This calculation is based on the lumped capacitance model and treats the cavity temperature like the simple RC low-pass filter with a break point at f ext. Temperature fluctuations above this frequency will be heavily filtered, see the example in figure Using the value of τ ext the cut off frequency for the sapphire cavity is Hz. Figure 2.10: Simulated thermal transfer function of a low-pass filter 22

39 2.4. THERMAL TRANSFER FUNCTION Heat Shield External Time Constant We can calculate the external time constant of the aluminium heat shield in a similar manner. The aluminium heat shields are also mounted on teflon supports, though these are larger than the ones supporting the sapphire cavity with a cross sectional area of 15 cm 2, average length 3 cm and a thermal resistance calculated at 40 K/W. The larger surface area of the heat shield and lower emissivity of machined aluminium (ε = 0.07) combine to give a radiative thermal resistance of 60 K/W. Adding the conductive and radiative thermal resistance we get a total resistance of 24 K/W. The heat shields were designed for a maximum heat capacity with minimum size, to increase the thermal time constant. The large thermal capacity of the heat shields (5230 J/K) gives an external time constant of seconds or a low pass filter with a cut-off frequency of Hz. With the aluminium heat shield we are able to verify the above calculations by measuring the steady state situation when the heat into the system Q is equal to the heat flow out of the system. The aluminium heat shield is at a constant temperature when there is a total heat input of W. The temperature difference between the aluminium heat shield and its surrounds was measured to be 1.14 K, so a calculation of: R = ΔT Q 2.19 gives a total thermal resistance of 30 K/W, which is in reasonable agreement with the calculated value of 24 K/W. Another method to validated the above calculations is to measure the response of the sapphire cavity to an external perturbation of temperature. In figure 2.11 the set point of the aluminium heat shield is changed and we can see the response to this change in heat input. As expected it is an exponential increase in temperature of the form e t/τ and we have fitted the appropriate exponential. The time constant is found to be seconds ( 38 hours, f ext = Hz) and this is in reasonable agreement with the calculation presented in section The aluminium 23

40 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY heat shield where the temperature change took place is immediately surrounding the cavity and there will be further filtering from the outside environment through the other thermal layers and the temperature control scheme outlined in the previous section. Figure 2.11: Sapphire cavity response to a deliberate change in temperature of the aluminium heat shield. The blue line is an exponential fit with time constant seconds Heat Shield Internal Time Constant The internal time constant for heat flow within a body is not considered in the lumped capacitance model [Dratler 74]. In this experiment it must be taken into account due to the long external time constants of the aluminium heat shields. The internal time constant of the heat shields is determined experimentally and a simulation of heat flow within a large object is used to further the analysis. To determine the internal time constant of the heat shield a measurement of the thermal transfer function was undertaken from to 0.3 Hz and is shown in figure A sinusoidal voltage was applied to the heater and the thermal response of the heat shield was recorded over many cycles. The frequency of the sine wave is varied to obtain sufficient data points, and the phase and amplitude of the response gives the transfer function. The thermal transfer function of the large aluminium heat shield illustrates some 24

41 2.4. THERMAL TRANSFER FUNCTION Figure 2.12: Thermal transfer function of the aluminium heat shield measured using the method of sines. key elements of heat flow within a macroscopic room temperature object. The flat region from to Hz is an unexpected feature that is not often reported in other temperature control applications. The high levels of thermal isolation reveals behavior not seem in more thermally coupled systems where the external time constant is at much shorter time scales. The flat region is caused by the aluminium heat shield acting like a thermal bath for the heater-thermistor combination, which gives rise to another break-point. It is analogous to the external time constant but instead of describing the flow of heat between the heat shield and the environment it relates the flow of heat between the thermistor/heater combination and the heat shield. It is only on time scales longer than it takes the heat shield to fill up with thermal energy that it no longer resembles a thermal bath and the transfer function begins to slope upwards to the f ext break-point. The parameters determining how long it takes to saturate an object with thermal energy is the internal thermal resistance and the thermal capacity of the object. The thermal resistance between the heat source and the object limits the rate of heat transfer compared to the total thermal capacity of the object. The time constant in this case is given by: τ int = R int C sh

42 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY Where R int is the thermal resistance between the heater/thermistor and the heat shield. Using the the thermal resistance of 0.35 K/W from the measured transfer function (figure 2.12) and the previously calculated total thermal capacity of the heat shield C sh = 940 J/K, the internal time constant is 1560 s and the break-point is expected at Hz. Examining figure 2.12, the thermal transfer function is increasing below this frequency suggesting the internal flow of heat is restricted by the heater/thermistor thermal resistance to the heat shield. However, the lumped capacitance model we have used here is not always valid for room temperature macroscopic objects since it does not describe the spatial flow of heat within bodies. There is another regime in which the time taken to fill an object with heat is dominated by the diffusion of heat to the edges of the object. In this case the limiting factor is, (to first approximation [Fowkes 94]), the time constant: τ int = L2 π 2 κ 2.21 where L is the largest distance from the heat source to the edges of the object and κ is the thermal diffusivity of the material. To illustrate this point the heat flow in a 30 cm aluminium cube was modeled using Mathematica. The 3D heat equation: ( ) T 2 t = κ T x + 2 T 2 y + 2 T + Q(t) 2 z was solved using symmetric boundary conditions with a region of heat input Q(t) at the top-centre of the cube and constant heat loss at the surfaces, see figure The thermal transfer function was determined in a similar method to the previous experiment, the temperature was recorded at a point 2 cm from the heat source and the heat input was varied sinusoidally at decreasing frequencies. The thermal transfer function is shown in figure The simulated thermal transfer function shows the expected break points f ext and f int plus another f th that will be explained shortly. Using equation 2.21 to calculate f int as we see that the break-point occurs where expected. It makes sense that the simulation is dominated by the time taken for the thermal energy to diffuse through the object since the model contains no thermal resistance between the heat input and the rest of the object to hinder heat flow. In the actual heat 26

43 2.4. THERMAL TRANSFER FUNCTION Figure 2.13: The aluminum cube used in the heat flow simulation. Q(t) is the heat input and the temperature is measured at T(t) Figure 2.14: Simulated transfer function of a 30 cm aluminium cube. (The phase at frequencies >0.1 Hz is wrapped back to within 0 and 180 ). shield we may take as the longest dimension the distance around the hollow cylinder to the opposite side, 23.5 cm. Again using equation 2.21 the internal time constant for heat diffusion is 67 seconds, 23 times faster than the time constant associated with heat input. The existence of an internal break point, whether it is associated with heat transfer or diffusion, is a unique result with implications for temperature control of macroscopic objects at room temperature. 27

44 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY The last break point in the thermal transfer function is f th which is related to the time constant of the thermistor and the time it takes for the heat to reach the thermistor from the heater. The break-point frequency we expect here is 0.15 Hz from the thermistor time constant [Dratler 74] but what is more important from a control viewpoint is the time delay as the heat flows from the heater to the thermistor. This causes a phase lag (see figures 2.12 and 2.14) and ultimately limits the bandwidth of any temperature controller. From the previous measurements, the external time constant of the sapphire cavity is seconds and external temperature fluctuations faster than this will be reduced. The external time constant of the heat shields is seconds and the internal time constant for the diffusion of heat in the shield is 67 seconds. Therefore the heat shield will reach internal equilibrium 2000 times faster than it will come to equilibrium with the external environment. This gives the heat shield a uniform temperature that can be stabilized by single point control. The internal time constant of 1560 seconds associated with the thermal resistance from the heater/thermistor combination to the heat shield limits the ability of the control system to respond to temperature changes of the heat shield. In this experiment this was not seen as a major drawback as there are multiple stages of active temperature control and passive filtering between the heat shield and the external environment which will heavily filter fluctuations in the temperature surrounding the vacuum system. 2.5 Temperature Stability The purpose of the temperature control system and the thermal isolation of the sapphire cavities is to stabilize the temperature of Fabry-Perot cavities. The temperature of the sapphire spacers is measured by a 10 kω thermistor thermally bonded to the mirror clamps in a four-wire configuration. A four-wire resistance measurement is more accurate than a two-wire measurement since the resistance of the leads is not included [Moore 83]. The data was acquired by a DVM (Agilent 34401A) with 28

45 2.5. TEMPERATURE STABILITY 6 digit accuracy and a 2 second gate time. Figure 2.15 shows the temperature variations over several days from one of the cavities. The temperature variations for a similar time period from the previous temperature control setup [Dawkins 07] are included to highlight the improvement in temperature stability. Figure 2.15: Comparison of the old (red) and new (blue) temperature control over similar time scales. The previous temperature stability showed significant variations on daily time scale that are absent in the upgraded system. Reduction of these systematic fluctuations was a major goal of the improved temperature control system because the original intention of using the sapphire Fabry-Perot cavity as a frequency reference for a test of Lorentz Invariance required the minimal frequency modulations with a 24 hour period. To characterize the temperature stability over a wide range of time scales we used the Square Root Alan Variance (SRAV) [Allan 74]. Originally developed to characterize the frequency stability of atomic clocks the SRAV is the square root of the total variance between adjacent data points in a data set. To lengthen the integration time τ the data is averaged over an increasing number of points to generate a measure of the expected variations at different time scales. A plot of the temperature SRAV from a typical data set is shown in figure 2.16 and the noise floor of the thermistor is evident in the trace flattening out at 15 μk. We can infer the temperature fluctuations below this level from the measured resonance 29

46 CHAPTER 2. TEMPERATURE CONTROL OF A FABRY-PEROT CAVITY frequency variations of the sapphire Fabry-Perot cavities. On time scales of a day Figure 2.16: Square root Allan variation of cavity temperature fluctuations. The red line is temperature data from the 4-wire thermistor on the sapphire cavity, the blue line is the temperature fluctuations inferred from the resonance frequency fluctuations. This shows the high sensitivity of the sapphire Fabry-Perot cavities to temperature and the thermal filtering of cavities themselves. the temperature of the sapphire spacer fluctuates 300 μk which implies an average drift rate of just 3 nk/s. This is 2 orders of magnitude better than the previously measured temperature drift of the sapphire cavity. Using the known temperature dependence of the cavity resonant frequency the expected frequency drift due to temperature variations is < 4 Hz/s, which is makes the sapphire cavities useful as an optical frequency reference in the laboratory (see section 4.7). 2.6 Conclusions Although the temperature stability of the sapphire cavities was vastly improved the performance of the experiment was insufficient for the original intended purpose, as a reference cavity for a test of Lorentz invariance. To reduce the drift rates to levels acceptable for a test of Lorentz Invariance the temperature stability would need to be improved another 2 orders of magnitude to μk over 24 hours. Temperature control of such a large object at an exceptionally high level for long time scales is 30

47 2.6. CONCLUSIONS inherently difficult [Unni 03] and this highlights the major disadvantage of using a cavity spacer with a non-zero coefficient of thermal expansion. As an optical frequency reference the sapphire cavities are extremely sensitive to any temperature variations, random fluctuations of the laboratory temperature may be causing small thermal gradients in the vacuum system. Thermal gradients can bypass the single point control generating the random temperature fluctuations seen by the sapphire cavity. Alternatively the temperature controller itself may be disrupted by changes in the laboratory environment giving rise to long term fluctuations. Either source of variation is difficult to identify and eliminate. Ultimately the stringent requirement of temperature stability was rendered unnecessary by advances made in the method of testing LI, covered in later chapters. However the large reduction in temperature drift of the cavities made them significantly better optical frequency references and eliminated the need for frequency control systems with massive ranges (10s of MHz) to accommodate for the large variations in resonance frequency. This has enabled the introduction and testing of more sophisticated frequency locking techniques. 31

48

49 If everything seems under control you re not going fast enough Mario Andretti ( present) 3 Frequency Control of a Laser Chapter Overview In this project the frequency stabilization of the laser with the Fabry-Perot resonators was achieved by the Pound-Drever-Hall locking scheme and the technique is detailed in the first section, with a review of basic control theory in the second. The optical setup and the specifics of the frequency locking scheme are also shown. The laser is locked to one resonator through a piezo-electric transducer which directly modulates the laser frequency, while an acousto-optic modulator is used to lock to the other resonator. The use of an acousto-optic modulator in the locking scheme provides the frequency difference of the resonators without the need for an optical beat note, and the use of acousto-optic modulators for fast frequency control is characterized. 3.1 Frequency Stabilization with an Optical Cavity In order to stabilize the frequency of laser to an optical resonator it is necessary to obtain a signal that can be used to control the frequency of the laser. In this experiment we use the classic Pound-Drever-Hall (PDH) laser frequency stabilization technique [Drever 83] to lock the frequency of the laser to the resonance frequency of the stable Fabry-Perot optical cavities. The PDH scheme requires the phase or frequency of the incident laser to be modulated and by a suitable de-modulation of the reflected light an error signal is generated that is proportional to the laser de-tuning from resonance. This error signal is used to stabilize the frequency of the 33

50 CHAPTER 3. FREQUENCY CONTROL OF A LASER laser. We base our analysis of the PDH technique on that described by [Riehle 04, Black 01]. An incident laser beam is phase modulated at frequency Ω Ẽ mod = E 0 e i(ωt+βsinωt) 3.1 = e iωt [cos(βsin(ωt)) + isin(βsin(ωt))] 3.2 E 0 is the amplitude and ω is the angular frequency of the electric field and β is the modulation depth. The modulated beam can be thought of as consisting of a carrier at ω and two sidebands at ω ± Ω. This is evident if we expand the previous expression in terms of Bessel functions of the first kind, J n (x). The expansion is: cos(x sin(θ)) = J 0 (x)+2j 2 (x)cos(2θ)+... sin(x sin(θ)) = 2J 1 (x)sin(θ)+2j 3 (x)sin(3θ) If we only take the first terms in each series, after some algebraic manipulation the modulated beam then becomes Ẽ mod E 0 e iωt [J 0 (β)+2j 1 (β)sin(ωt)]e iωt = E 0 [J 0 (β)e iω + J 1 (β)e i(ω+ω) J 1 (β)e i(ω Ω) ] 3.5 and equation 3.5 explicitly shows the sidebands at ω ± Ω. Going back to equation 2.2, the reflected amplitude Ẽref will be Ẽ ref = F (ω)ẽmod = E 0 [F (ω)j 0 (β)e iω +F (ω+ω)j 1 (β)e i(ω+ω) F (ω Ω)J 1 (β)e i(ω Ω) ] 3.6 In a real experiment we measure the power of the reflected beam P ref = E ref 2, given by: P ref = P c F (ω) 2 +P s [ F (ω +Ω) 2 + F (ω Ω) 2 ] +2 P c P s Re[F (ω)f (ω +Ω) F (ω)f (ω Ω)]cos(Ωt) +2 P c P s Im[F (ω)f (ω +Ω) F (ω)f (ω Ω)]sin(Ωt) +(cos(2ωt) and sin(2ωt) terms)

51 3.1. FREQUENCY STABILIZATION WITH AN OPTICAL CAVITY where P c = E 0 J 0 (β) 2, the power in the carrier and P s = E 0 J 1 (β) 2 is the power in the sidebands. For β<1negligible power goes into the higher order sidebands so P c +2P s P 0. The terms of equation 3.7 can be interpreted in terms of the carrier and sidebands of the incident modulated beam: the first term is just the reflected carrier; the second term is the reflected sidebands; the third and fourth terms modulated, at Ω, arise from the interference of the carrier with the sidebands and the 2Ω terms are the two sidebands interfering with each other. We are interested in the third and fourth terms at frequency Ω because they contain information about the phase of the reflected sidebands relative to the carrier. The two sidebands have a well defined phase relationship with the carrier but are separated by the modulation frequency. On reflection from the cavity the phase of the components of the reflected beam are modified and the phase difference between the sidebands and carrier after reflection allow determination of the frequency of the laser relative to the optical cavity resonance. (a) Magnitude squared F (ω) 2 (b) Phase of F (ω) Figure 3.1: Behavior of the reflection coefficient near resonance for an optimally coupled cavity To form a DC signal suitable for a control loop a frequency mixer can be used to de-modulate the terms at frequency Ω in equation 3.7 to DC. A low-pass filter can then be used to discard the other terms. An ideal mixer essentially multiplies two signals together to generate a signal at the sum and difference frequencies. 35

52 CHAPTER 3. FREQUENCY CONTROL OF A LASER Mathematically this is: A 1 cos(ω 1 t) A 2 cos(ω 2 t)= A 1A 2 [cos(ω 1 ω 2 )+cos(ω 1 + ω 2 )] If ω 1 = ω 2 and a low pass filter follows the output of the mixer then the DC signal out of the mixer is proportional to the product of the amplitudes A 1 A 2.Ifwekeep the amplitude of the second input to the mixer constant then the output will be proportional to the amplitude A 1. In this way we can use a mixer to isolate the Ω varying terms in equation 3.7 to obtain the required error signal. The frequency of the modulation determines whether we isolate the sin(ω t) or cos(ω t). The determining factor is whether the function: F (ω)f (ω +Ω) F (ω)f (ω Ω) 3.9 present in equation 3.7, is real or imaginary. If Ω is large compared to the cavity resonance ν cav then the sidebands are reflected and the reflection coefficient F(ω ± Ω) 1 so equation 3.9 is F (ω) F (ω) = 2 Im[F (ω)] 3.10 which is purely imaginary. If the modulation frequency is small compared to the the cavity resonance ν cav then we can re-write equation 3.9 as [ F (ω)f (ω +Ω) Ω + F (ω)f ] (ω Ω) Ω Ω [( Ω F (ω) d ) ( dω F (ω) + F (ω) d ) ] dω F (ω) [ = 2Ω Re F (ω) d ] dω F (ω) 3.11 which is purely real. In the case where Ω ν cav (as is the case here) both signals will be present and we can adjust the phase of the reference arm of the mixer to demodulate the required quadrature, see figure 3.2. In both cases we are detecting the interference of the sidebands with the light circulating in the cavity. If the modulation frequency is slow relative to the cavity 36

53 3.1. FREQUENCY STABILIZATION WITH AN OPTICAL CAVITY (a) Ω = = Δν cav /10 (b) Ω = 0.05 = 5Δν cav Figure 3.2: The error signals after demodulation for slow (a) and fast (b) frequency modulation (relative to the cavity bandwidth). The two traces on each graph are the two quadratures of the signal. bandwidth then the incident beam is in equilibrium with the light circulating in the cavity. In this case we can consider the sidebands as reflected with modified phase from the circulating beam and this phase difference tells us which side of the resonance we are on. If the modulation is faster than the cavity bandwidth we are interfering the promptly reflected sidebands with the leakage field from the cavity which is a time averaged version of the laser frequency. The feature common to both error curves in figure 3.2 is a signal varying essentially linearly with frequency around the resonant frequency. This forms the basis of the control loop used to stabilize the frequency of the laser to the resonant frequency of the optical cavity. The sensitivity of the system to fluctuations in frequency is the derivative of the error signal at the resonance frequency (the frequency discriminator) and this generally determines the performance of the frequency control system. A plot of the derivative (taken numerically using Mathematica) is shown in figure 3.3 as a function of modulation frequency, with constant modulation depth. The frequency discriminator slope has a dependance on the modulation depth β. It is proportional to P c P s so,ifweassumethatp c +2P s P 0 then the frequency discriminator slope traces out the semi-circle shown in figure 3.4 as a function of the power in the sidebands. The maximum frequency discriminator slope occurs at P s /P s 0.5 and thus we adjusted the modulation amplitude for this situation. 37

54 CHAPTER 3. FREQUENCY CONTROL OF A LASER Figure 3.3: The absolute value of the derivative at resonance for the two error signal quadratures (red is cosine, blue is sine) shown in figure 3.2 as a function of modulation frequency with constant modulation depth. The cavity has a bandwidth of Δν cav =0.01 Figure 3.4: The frequency discriminator slope as a function of sideband power P s /P c The reflection coefficient F(ω) plays a defining role in the derivative of the error signal near resonance and F(ω) is dependent on the parameters of the cavity. To maximize the frequency discriminator of the cavity the contrast C should be high and the cavity bandwidth ν cav should be narrow, or in other words the cavity should have a high finesse F - see equation 2.5. For an optical cavity of a given length this gives the contrast multiplied by the cavity finesse, C F as an approximate figure of merit when comparing optical resonators. With a linear error signal a control loop can be engaged to lock the frequency of the laser to the cavity resonance and 38

55 3.2. CONTROL LOOPS suppress fluctuations in the the frequency of the laser. 3.2 Control Loops Before we commence with the details of the experiment, a basic review of control theory will be useful. The signals in the system will be represented by their Laplace transform L{f(t)} = F (s) which is an integral transform with complex argument, s, formally defined as: F (s) =L{f(t)} = 0 e st f(t)dt 3.12 The ratio of the Laplace transform of the output to input gives the transfer function of a device which characterizes its operation. The argument s is a complex variable: it contains information on the amplitude and phase of the output relative to the input and is used extensively in control theory. The idea of a control loop is to stabilize the output of a system with respect to a steady reference. The most simple control loop is shown in figure 3.5. Figure 3.5: A simple control loop with stable input X(s), gain stage G(s), plant P(s) and output Y(s) Here the input to the loop is X(s) and the output is Y(s). The difference between them is amplified by the control system G(s) and fed back into the plant P(s) - the device that is under control. In our experiment the resonance of the optical cavity is the stable reference X(s) and Y(s) is the output of the frequency stabilized laser. The PDH locking scheme determines the difference between the laser frequency and the resonance of the optical resonator and this error signal gets fed into the servo control 39

56 CHAPTER 3. FREQUENCY CONTROL OF A LASER electronics. The control electronics are equivalent to the gain transfer function G(s) in figure 3.5 and the device controlling the frequency of the laser, the laser itself and the detection electronics can be considered the plant, P(s). The input to the control system is called the error signal, Err(s), and the output of the control system, Cor(s), is the correction required to stabilize the output. These naming conventions will be used throughout this thesis. Analyzing the transfer function around the loop we get: Y (s) = P (s)cor(s) Cor(s) = G(s)(X(s) Y (s)) Y (s) = G(s)P (s) 1+G(s)P (s) X(s) 3.13 Equation 3.13 is often called the closed loop gain of the system and G(s)P(s) the open loop gain. As the gain of the controller G(s) is increased and G(s)P(s) then equation 3.13 X(s) and the output of the loop matches the input. Equivalently the error: Err(s) = 1 1+G(s)P (s) X(s) 3.14 approaches zero. The frequency at which the open loop gain drops below one, termed the unity gain frequency, is the frequency at which the loop is no longer effectively suppressing noise. This essentially defines the bandwidth of the control loop. The loop becomes unstable if the G(s)P(s) approaches -1 and there is amplification of noise in this situation. This occurs when the system has unity gain P (s)g(s) 1 and the phase delay reaches 180 degrees. All real systems have delay at some frequency as the reciprocal of the response time of the device becomes appreciable compared to the input frequency. If the phase lag at the unity gain point is large enough then the system will oscillate at a frequency close to bandwidth of the control loop. For most systems, stability is assured if the open loop gain is less than unity as the phase delay nears -180 degrees. At high open loop gain close to the threshold of oscillation the control loop will be stable, but there may be some amplification of noise at the bandwidth of the loop - a servo bump A common form of control is the PI (Proportional Integral) controller. transfer function is designed to account for the phase delay seen at high enough 40 The

57 3.2. CONTROL LOOPS frequencies and still have plenty of low frequency gain. An example transfer function of a PI controller with a break point at 10 khz is shown in 3.6. An integrator has the desirable quality of increasing gain at low frequencies but exhibits a 90 degree phase lag which can combine with the phase delay of the plant to create an unstable loop at high frequency. This is overcome by making the transfer function proportional above some break point frequency and adjusting the open loop gain to avoid oscillations. (a)gainofapicontroller (b) Phase of a PI controller Figure 3.6: An example transfer function of a PI controller with a break point at 10 khz showing the 90 phase delay of an integrator In a real system noise will be present in the system and we will consider the simple case of noise at the input of the control loop N i (s) and noise at the output N o (s), as shown in figure 3.7. In this system the loop equations become: Figure 3.7: A simple control loop input noise N i (s) and output noise N o (s) 41

58 CHAPTER 3. FREQUENCY CONTROL OF A LASER Y (s) = Err(s) = G(s)P (s) 1+G(s)P (s) [X(s)+N 1 i(s)] + 1+G(s)P (s) N o(s) 1 1+G(s)P (s) [X(s)+N i(s) N o (s)] In this case as the gain G(s)P(s) the second term in equation 3.15 goes to zero and the output noise is suppressed. The second term approaches X(s) +N i (s) so the stability of the output is limited by the input noise. However we can see from equation 3.16 that the error signal Err(s) still goes to zero so a measurement of inloop noise it not representative of the output noise which may be much higher. This highlights the danger of using in-loop measurements to determine the performance of a control loop. 3.3 Initial Optical Setup The laser used throughout this thesis is a Light Wave Electronics model 124. It is a Nd:YAG diode-pumped, non-planar ring laser with a nominal wavelength of 1064 nm. The laser has a specified linewidth of 10 khz in 1 ms of integration time and a drift rate of 50 MHz/hour, with a maximum laser power output of 50 mw. The laser has two methods of controlling the output frequency: For fast frequency tuning there is a piezo-electric transducer bonded to the laser crystal and for slow broadband frequency tuning, the temperature of the laser crystal is varied. The free-running frequency fluctuations of the laser output, shown in figure 3.8, were measured from the PDH error signal whilst the cavity is on resonance and the frequency control is unlocked. We aim to reduce these fluctuations as much as possible by frequency locking the laser to a resonance of a stable Fabry-Perot cavity. For the output of the laser to be well coupled into a Fabry-Perot resonator it s beam shape must be spatially matched to the fundamental spatial mode of the cavity. The beam straight out of the laser is slightly astigmatic and to correct this two cylindrical lenses were used. A 200 mm focal length lens orientated vertically and a 300 mm focal length lens orientated horizontally reduce the astigmatism. A 500 mm focal length lens is placed 500 mm away from the cylindrical lens, it was 42

59 3.3. INITIAL OPTICAL SETUP Figure 3.8: The power spectral density of the free-running fluctuations in the laser frequency. calculated from Gaussian optics that this produces a circular and well collimated beam with a 1 mm spot size and this was confirmed experimentally. To prevent back-reflected beams from interfering with the operation of the laser there is an optical isolator immediately at the output if the laser and a polarizer to maintain a constant polarization Figure 3.9: Simplified mode-matching optics. The beam astigmatism is corrected by the two cylindrical lenses (1) and the beam is collimated by (2). Lens (3) is chosen to match the waist size of the fundamental mode inside the Fabry-Perot cavity (4) To mode-match the light into the Fabry-Perot cavity the incident beam should spatially overlap the fundamental spatial mode. Both Fabry-Perot cavities have one 43

60 CHAPTER 3. FREQUENCY CONTROL OF A LASER flat mirror and one curved mirror so the resonating beam will have a waist at the flat mirror, and both cavities have been orientated so this is the incident mirror. From [Siegman 86] the waist at the flat mirror is: ω0 2 = λ L(R L) π 3.17 where λ is the wavelength of the light, L is the cavity length and R is the radius of curvature of the curved mirror. The required waist size for OFR 1 (R=500 mm) is 278 μm andforofr 2 (R=222 mm), 188 μm. A single mode matching lens which produces a waist of the correct size is placed approximately one focal length away from the front face of the Fabry-Perot cavity, and the process is quite forgiving. Calculations indicated for OFR 1 a 500 mm lens, and for OFR 2 a 300 mm lens would be suitable. A camera viewing the transmission through the cavity confirmed the fundamental mode was excited. Since PDH locking requires the reflected beam a simple half-wave plate (HWP) polarizing beam splitter (PBS) quarter-wave plate (QWP) combination (see figure 3.10) was used to isolate the reflected beam onto the control photodiodes. In the initial stages of the project the photodiodes used for the frequency control were silicon based New Focus model Figure 3.10: Reflected beam isolation. HWP -half wave plate, PBS - polarizing beam splitter, QWP - quarter wave plate. To measure the frequency stability of an optical reference it must be compared to a device with similar performance, and a sensible way to achieve this is with two Fabry-Perot cavities. Each cavity must have an independent locking and frequency control system but we use one laser, therefore it was necessary to develop a system of locking to both cavities using a single laser. This was achieved by using an Acousto- Optic Modulator (AOM) for frequency control. In this initial optical setup the laser 44

61 3.3. INITIAL OPTICAL SETUP frequency is locked to OFR 1, actuated by the piezo-electric actuator (PZT) on the laser crystal. Before OFR 1 some of the laser beam is redirected through an AOM that is used for to vary the optical frequency to enable a lock to OFR 2. An outline of the system is shown in figure 3.11 and the details of the scheme are covered in the following sections. The optical properties of the two Fabry-Perot cavities are repeated in Table 3.1 Table 3.1: The cavity parameters from [Dawkins 07], confirmed experimentally. OFR 1 OFR 2 FSR (ν FSR ) 1.02 GHz GHz Bandwidth (ν cav ) 840 khz 177 khz Finesse (F) Contrast (C) Waist Size 278 μm 188 μm Figure 3.11: The Initial frequency control setup. The laser is frequency locked to OFR 1 through the piezo-electric actuator on the laser crystal. The laser is then split into another arm which is frequency locked to OFR 1 through the acousto-optic modulator (AOM) 45

62 CHAPTER 3. FREQUENCY CONTROL OF A LASER 3.4 OFR 1 Frequency Lock - Piezo We produce the required phase modulation of the laser by using the piezo-electric transducer on the laser to create frequency modulation [Cantatore 95]. It was found that modulation by piezo-electric actuator produces significantly less unwanted residual amplitude modulation and is more stable than can be achieved using an electro-optic modulator. If the laser is frequency modulated the at Ω Rad/s then the laser angular frequency is: ω(t) =ω 0 + A cos(ωt) 3.18 and from the definition of phase as the time integral of the angular frequency we get t φ(t) = [ω 0 + A cos(ωt)] dt = ω 0 t + A Ω sin(ωt) 3.20 The frequency modulation is created at khz from an Agilent 33120A function generator that is phase locked to the 10 MHz reference signal from a KVARZ H-Maser. The modulation frequency required is well above the bandwidth of the piezo-electric actuator and to get enough power in the sidebands we must modulate at a resonant frequency of the piezoelectric actuator. The light reflected from the cavity is detected by a New Focus model 2031 large area photodiode on the medium gain setting. The reflected beam signal is passed through a band-pass filter centered around the modulation frequency and amplified by a mini-circuits ZFL-500LN low noise amplifier. The PDH signal is then mixed down to DC by a mini-circuits ZAD-3B frequency mixer. The LO port of the mixer is driven by another Agilent 33120A function generator phase locked to the 10 MHz reference signal and putting out a khz signal at 5 dbm. The phase of the second function generator is adjusted to maximize the slope of discriminator. The output of the mixer is passed through a 100 khz low pass filter to reject high frequency mixing products and into a Precision Photonics LB1005 Lock Box. The Lock Box is a high speed servo controller with 46

63 3.4. OFR 1 FREQUENCY LOCK - PIEZO flexibility in the gain and bandwidth of the control electronics. The DC output of the Lock Box is summed to the laser frequency modulation signal with a minicircuits ZFBT-6GW bias-tee to control the frequency of the laser through the PZT actuator. The locking scheme is shown schematically in figure In terms of the filtering, amplification and de-modulating before the Lock Box, the same scheme is used to provide the error signal for OFR 2. Figure 3.12: The frequency locking scheme for OFR 1 The gain and bandwidth settings needed on the Lock Box servo controller are determined by the transfer function of the PZT actuator, the photodiode and the demodulation system. As it is a mechanical device the PZT actuator on the laser crystal is the slowest component and will limit the bandwidth of the laser frequency controller. The transfer function is shown in figure To measure the transfer function the in-built source of a spectrum analyzer is used to produce a swept sine wave to drive the piezo-electric actuator. The measurement is taken while the laser is on resonance but unlocked, and the transfer function is determined by the PDH error signal. The Lock Box was adjusted to have a PI filter with a break point at 10 khz to account for the bandwidth of the PZT actuator. The PZT actuator on the laser crystal has a limited range and slow drifts in the temperature of the Fabry-Perot cavity can cause the laser to unlock once that range has been exceeded. To prevent 47

64 CHAPTER 3. FREQUENCY CONTROL OF A LASER Figure 3.13: The transfer function of the PZT actuator used for frequency control for OFR 1. Note the low-q resonance at 10 khz. this an integrator with a very slow time constant (> 1 second) was used to servo the output of the Lock Box to zero by steering the temperature of the laser, keeping frequency control system locked indefinitely. With the control system for OFR 1 engaged the level of frequency fluctuations in laser were reduced dramatically. This enables an AOM to be used for frequency corrections in the other arm of the laser system locked to OFR OFR 2 Frequency Lock - AOM An acousto-optic modulator operates by the Bragg diffraction of an incident laser beam from a moving acoustic wavefront. A piezo-electric transducer is used to generate acoustic waves at RF frequencies in a small piece of photoeleastic material [Young 81]. The incoming laser photons are scattered off the acoustic phonons in the material with a corresponding change in energy and momentum. If the incoming light has frequency ω with wave vector k = ωn/c, and the phonon has frequency Ω A and wave vector κ =Ω A /v s,wherev s is the speed of sound in the material, the resulting diffracted beam after first order diffraction has: ω d = ω ± Ω A 3.21 k d = k ± κ

65 3.5. OFR 2 FREQUENCY LOCK - AOM The phonon can be absorbed or emitted depending on the angle of the incoming beam and this dictates the sign in the above equations, the phonon has been absorbed in figure 3.14 Figure 3.14: Acousto-optic Modulator. The incoming laser of get frequency shifted by Bragg diffraction from the moving acoustic waves Using an AOM as a frequency modulator relies on equation the frequency of the diffracted beam gets shifted by the AOM frequency Ω A. By changing Ω A we can control the laser frequency to lock the laser to the Fabry-Perot cavity OFR 2. However there are features of frequency control with an AOM that must be dealt with to ensure a high performing lock. by: The Bragg diffraction angle θ Bg changes as the AOM driving frequency is changed sin(θ Bg )= κ 2k i 3.23 so as the drive frequency is changed to control the laser frequency the beam output angle from the AOM will vary. This is undesirable as it will change the alignment of the beam mode matched to the Fabry-Perot cavity. To overcome this the AOM is used in the double-pass configuration [Donley 05]. The diffracted beam is passed through a lens and mirror cat s eye to retro-reflect the beam back into the AOM, see figure The ray-matrix for a cat s eye retroreflector with lens - mirror spacing d and mirror radius of curvature R is [Snyder 75]: 1 2d2 2d R

66 CHAPTER 3. FREQUENCY CONTROL OF A LASER If the beam leaving the AOM has position from the optical axis r i and slope r i then from the paraxial ray approximation the beam will return to the AOM with properties r o = 1 L 1 2d2 2d R 1 L r i r o r i 3.25 = r i +2d(d L d2 R )r i r i 3.26 in our case the mirror is flat (R= ) and d = L. Equation 3.26 becomes [ r i, r i], and the cat s eye compensates for the deflection of the diffracted beam as the frequency of the AOM is varied. After the second pass the beam maps back onto the incident beam and the output of the AOM can be separated from the input by putting a quarter wave plate in the cat s eye and using the change in polarization from a double pass of a quarter wave plate to isolate the input from the output with a polarizing beam splitter. The non-frequency shifted beam is blocked after the cat s eye and on the second pass the non-frequency shifted beam diverges from the frequency shifted diffracted beam and was blocked with an iris. Since the beam is passed through the AOM twice, the frequency is shifted by double the AOM driving frequency. The cat s eye also serves to keep the incident beam at the optimal Bragg angle θ Bg which allows a much larger frequency tuning range than with using just a mirror for a double pass configuration. Figure 3.15: Acousto-optic Modulator in the double pass configuration. The acousto-optic modulator we use is a Crystal Technologies model The crystal material is TeO 2 and the nominal driving frequency Ω A is 80 MHz 50

67 3.5. OFR 2 FREQUENCY LOCK - AOM provided by an Agilent N181A signal generator amplified by a mini-circuits ZHL-1-2W high power amplifier to 25 dbm. The signal generator is phase locked to the 10 MHz reference and has external inputs to Frequency Modulate (FM) and Amplitude Modulate (AM) the output signal. The FM input on the signal generator is used for frequency control by the OFR 2 control loop to lock the frequency of the beam to the resonance of OFR 2, see figure To obtain an error signal the same demodulation frequency of khz is used because the PDH frequency modulation from the lock on OFR 1 is present in the laser. Figure 3.16: frequency control. The Locking electronics for OFR 2 using a double pass AOM for The bandwidth of the lock in OFR 2 will be limited by the transit time for the acoustic waves in the AOM crystal to propagate from the piezo-electric transducer across the optical beam [Hobbs 09]. The speed of sound in TeO 2 is given by the manufacturer as m/s and the beam has been collimated to a 1 mm spot size positioned 3.7 mm from the piezo electric transducer, because of the aperture on the AOM. For this configuration the maximum response time of the AOM is 870 ns giving a bandwidth limit for the lock of 181 khz. The measured transfer function of the frequency control system for OFR 2 is shown in figure 3.17, the 100 khz filter before the lock box was left in place and this limits the transfer function of the system. 51

68 CHAPTER 3. FREQUENCY CONTROL OF A LASER Figure 3.17: The transfer function frequency control of OFR 2. The transfer function is dominated by a 100 khz low pass filter on the output of the mixer. 3.6 Measuring Frequency Stability Using an AOM for frequency control allows us to measure the frequency difference, and hence frequency stability of the two Fabry-Perot cavities without the need for an optical beat note. The laser is locked to OFR 1 through the piezo-electric transducer on the laser crystal so the light branching off into the second arm to the AOM is already frequency stabilized to OFR 1 at frequency ν 1. The double pass AOM shifts the frequency of the light by twice the AOM center frequency (a constant 80 MHz, ν 80MHz ) plus the correction signal to the FM on the signal generator ν corr, to keep the second arm of the laser locked to OFR 2 at frequency ν 2. Therefore the correction signal to the AOM is the difference in frequency between OFR 1 and OFR 2 with a constant 80 MHz offset. Diagrammatically and mathematically this is shown in figure The correction signal to the AOM is equal to the optical beat note between the cavities and can be measured in two ways. It can be measured directly by using a frequency counter to measure the 80 MHz sent directly to the AOM by a frequency counter, which for these experiments is an Agilent 53131A. For fast frequency measurements the correction voltage sent to the signal generator FM can be recorded by a spectrum analyzer and converted to frequency variations because 52

69 3.6. MEASURING FREQUENCY STABILITY Figure 3.18: The laser frequency in the two arms showing the correction to the AOM is equivalent to a beat note. ν 2 = ν 1 +2Ω A and Ω A = ν corr + ν 80MHz. ν 2 ν 1 =2(ν corr + ν 80MHz ) there is a well defined relationship between the input voltage and the frequency shift of the light by the AOM. To verify that the correction signal is equivalent to the frequency difference of the Fabry-Perot cavities we constructed an experiment to create an optical beat note between the two cavities, shown in figure A phase locked loop was used to measure the optical beat note and converted into frequency fluctuations, and the correction signal sent to the AOM is converted to frequency by the sensitivity of the frequency modulation port on the N181A signal generator. The power spectral density of frequency fluctuations between the two cavities is shown in figure 3.20 measured with the correction voltage and an optical beat note. By inferring the optical stability of the experiment by measuring the correction signal sent to the AOM we have done away with the need for complex optical systems and fast photodiodes to obtain a direct beat note between the lasers locked to the two resonators. Fast frequency fluctuations can also be measured easily without using a Phase Locked Loop (PLL) by directly measuring the correction signal sent to the AOM with a spectrum analyzer. However this measurement can be misleading if the control loop is not properly engaged. Using the AOM correction signal as a measure of frequency difference in the two Fabry-Perot cavities requires that the laser is properly locked to the two cavities. Without the AOM control loop engaged the correction signal is just the output noise of the Lock Box. There must be sufficient gain to ensure the control 53

70 CHAPTER 3. FREQUENCY CONTROL OF A LASER Figure 3.19: Experiment to measure the optical beat note. A phase locked loop measured the frequency fluctuations in the optical beat note. Figure 3.20: Frequency fluctuations from the correction voltage (blue) and an optical beat note (red). The correction signal drops outside the bandwidth of the frequency control loop, the optical beat note contains a servo bump from the phase locked loop used to measure fast frequency fluctuations. loop is suppressing noise and actively locking to the second cavity, otherwise an erroneously quiet signal will be measured. At all times it was ensured that the control loop had sufficient gain to ensure that the measurement of the AOM signal represented the inter-cavity frequency stability. 54

71 3.7. ACOUSTO-OPTIC MODULATORS FOR FREQUENCY CONTROL Figure 3.21: Using the AOM correction signal to determine the frequency stability of OFR 1 and OFR 2. The laser is locked to OFR 1 and the AOM shifts the frequency of the light by the amount needed to keep the other arm locked to OFR 2. Therefore the difference in frequency (the beat note) between the two cavities is the signal sent to the AOM. This can be measured two ways: directly by a frequency counter from the 80 MHz sent to the AOM or from the voltage sent to the signal generator by a spectrum analyzer. 3.7 Acousto-Optic Modulators for Frequency Control The limitations and performance of an acousto-optic modulator are covered in this section. To have confidence in using the correction signal sent to the AOM as a measure of frequency stability it is necessary to establish that the AOM does not impart intrinsic frequency fluctuations on the light. Such noise will be at the output of the control loop, so, from section 3.2 it will be suppressed by the control loop. However, output noise may create a false noise floor at the point in the control loop we are observing, the correction signal sent to the AOM. We will consider the noise sources in an AOM used for frequency control in the following sections Frequency Noise Floors The signal generator exhibits effective noise to the input of the analog FM control. This is converted to unwanted frequency fluctuations in the signal sent to the AOM 55

72 CHAPTER 3. FREQUENCY CONTROL OF A LASER by the voltage to frequency conversion of the FM input on the signal generator. The Agilent N181A signal generator used to drive the AOM has a user defined setting of sensitivity for analog modulation of the output frequency. To measure the noise at the FM input we took two identical signal generators at 80 MHz with FM on and amplified them (as is done in actual experiment, before the AOM) then attenuated the signal to a reasonable level and input both signal generators into a mixer. The output of the mixer was sent to a Lock Box and then to the FM input of a signal generator to form a control loop, see figure This PLL locks the phase of one signal generator to the other, and the output of the Lock Box is the required frequency corrections to maintain the lock. It is a system somewhat analogous to the optical setup and will reveal the noise floor of the AOM signal, which, if below the measured fluctuations in the actual optical experiment implies that the signal generators do not limit our measurement of the frequency difference between the two Fabry-Perot Cavities using the AOM signal. Figure 3.22: Measuring the input noise of the signal generators. One signal generator is phase locked to the other and the correction signal out of the lock box required to maintain the lock is equal to the amount of noise in both the signal generators. Three power spectral densities were taken with different sensitivity settings on the the signal generators in terms of the Hz/V of the FM modulation. The input noise was tested with 10 khz/v, 100 khz/v and 1 MHz/V and the results are shown in figure From the figure it can be seen that the frequency noise increases as the frequency deviation (Hz/V) is increased, which is consistent with a fixed amount of voltage noise converted to larger amounts of frequency noise as the conversion 56

73 3.7. ACOUSTO-OPTIC MODULATORS FOR FREQUENCY CONTROL Figure 3.23: The input noise of two Agilent N181A signal generator with various settings for frequency deviation of the FM input.- 10 khz/v (light blue), 100 khz/v (blue) and 1 MHz/V (dark blue). Increasing the amount of frequency deviation increases the frequency noise which is consistent with a constant voltage noise at the input to the FM modulation. The data has a servo bump at 300 khz. factor is increased. For the 100 khz/v setting the noise is below 1 Hz/ Hz at 1 khz and falls further to 0.1 Hz/ Hz at 10 khz, the frequency region of interest in the optical experiment. This implies that the noise level of the signal generators do not limit the measurement of the optical stability of the Fabry-Perot cavities at the 100 khz/v setting, which was used throughout this thesis. To ensure this was the case some measurements of the intra-cavity stability were taken with the 10 khz setting the optical cavity stability was identical to the higher setting AOM Frequency Stability Unwanted frequency fluctuations could be caused by the interaction of the laser with the propagating acoustic waves in the AOM crystal. The easiest and most comprehensive way to measure effects such as these is to compare the frequency of the light before and after it has been frequency shifted by the AOM in situ. This was done with a dual AOM setup (see section 4.3.1). The setup is shown in figure 3.24 and involves passing the light through 2 double pass AOMs and making an optical beat note between the frequency shifted and unshifted light. The first double pass 57

74 CHAPTER 3. FREQUENCY CONTROL OF A LASER AOM shifts the light by +160 MHz and the second shifts it -140 MHz to give a beat note at 20 MHz, with both AOMs driven by Agilent N181A signal generators at a constant frequency with the FM turned off. The beat note was detected by an EOT 3000A fast photodiode and a PLL was employed using an Agilent E4428C signal generator to measure fast frequency fluctuations. Figure 3.24: The experimental setup to measure frequency fluctuations caused by the AOMs. Any frequency fluctuations imparted on the light by the AOMs will be present in the beat note between the shifted and unshifted light and is measured by the PLL. A power spectral density of the frequency fluctuations in the light after passing through two double pass AOMs is shown in figure The total frequency noise imparted on the light by the two AOMs is less than 1 Hz/ Hz and consistent with the noise floor of the PLL, so the AOMs are not limiting the measurement of frequency stability Acoustic Phase Delay We have shown that passive noise sources in the AOM are below the frequency noise between the two cavities. Since we are using the AOMs for active frequency control there are some subtleties that we must consider to ensure the experiment is not limited by the AOMs. As the driving frequency sent to the AOM is varied to keep the laser locked to the cavity the relative phase of the acoustic wave changes. This is because the acoustic wave must propagate a distance through the AOM crystal 58

75 3.7. ACOUSTO-OPTIC MODULATORS FOR FREQUENCY CONTROL Figure 3.25: Frequency fluctuations imparted on the light by two double pass AOMs driven at constant frequency. Again there is a servo bump, this time 50 khz due to the presence of a low pass filter at the output of the mixer. to the laser beam and different frequencies will reach the laser with different phases [Hobbs 09]. As shown before, frequency and phase are interrelated so unwanted phase shifts caused by actively controlling the AOM frequency will be a noise floor in the experiment. Figure 3.26: Exaggerated diagram of propagation of acoustic waves in the AOM crystal. As acoustic waves with different frequencies travel distance, d, to the laser beam they accumulate different phases relative to the input signal at the PZT. To measure the acoustic phase delay the same setup was used as the previous experiment to measure the intrinsic frequency noise of the AOM. The laser is frequency shifted by two AOMs and a beat note is made with the shifted and unshifted light. The driving frequency of one AOM is alternated between 80 MHz and a higher frequency (up to 84 MHz), and the phase shift of the beat note caused by the increase in driving frequency was recorded. The other AOM was driven at a constant frequency. 59

76 CHAPTER 3. FREQUENCY CONTROL OF A LASER Figure 3.27: Experiment to measure the acoustic phase delay of the AOM. The driving frequency of one of the AOMs was switched between 80 and 80+δ MHz and the phase shift of the beat note was measured. The measured acoustic phase delay for the AOM is shown in figure 3.28 as a function of driving frequency. The gradient of the line will determine the amount of unwanted phase shift introduced into the laser by changes in the AOM driving frequency, and from the graph it is measured to be Rad/Hz. Figure 3.28: Acoustic phase shift as a function of AOM driving frequency. The measured values are the dots and the line is a fit with a gradient of 5.5 Rad/Hz By simple consideration of the phase shift produced by waves of different frequency propagating in the AOM crystal it can be shown that the phase shift expect at distance, d, from the source of the waves is equal to: θ (Rad/Hz) = 2πd v s 3.27 and from this equation the beam is found to be 3.7 mm from the PZT on the AOM crystal, which is consistent with the observed optical setup. Multiplying the 60

77 3.7. ACOUSTO-OPTIC MODULATORS FOR FREQUENCY CONTROL acoustic phase delay by the frequency modulation sent to the AOM gives the amount of unwanted phase shift which can be converted back to frequency fluctuations. Such a graph is shown in figure 3.29 and shows the frequency noise well below 0.1 Hz/ Hz at the frequencies of interest. Figure 3.29: Unwanted frequency noise caused by the acoustic phase delay in the AOM crystal, inferred form the measured acoustic phase delay and the frequency modulation sent to the AOM Induced Amplitude Modulation Active frequency control by the AOM may introduce errors in the locking system through intensity fluctuations. The intensity of the frequency shifted beam out of the AOM varies slightly with frequency, see figure The PDH locking scheme should be insensitive to intensity fluctuations but small offsets in the demodulation system can induce sensitivity so the following experiment was carried out: The AOM driving frequency was deliberately modulated at a known level of frequency deviation (much larger than the correction signal sent to the AOM by the control loop) at a particular frequency. Then the amplitude modulation of the light at that frequency was recorded and this was repeated numerous times from 400 Hz to 20 khz, giving a measure of the Relative Intensity Noise (RIN) in the laser introduced by changes in the AOM driving frequency over a range of frequencies, see figure

78 CHAPTER 3. FREQUENCY CONTROL OF A LASER (a). To generate AM in the laser the RF power to the AOM can be varied. The efficiency of the AOM is proportional to the driving power so the intensity of the frequency shifted beam out the AOM can be changed by varying the input power from the signal generator using the built-in AM modulation. This was used to generate AM in the AOM at a particular frequency and the corresponding frequency modulation in the frequency of the two cavities was measured, see figure 3.31 (b). After this process we have, from the first measurement, a way to convert FM in the AOM to AM in the laser and, from the second measurement, a way to convert AM in the laser into unwanted FM in the frequency lock. Figure 3.30: Normalized optical output intensity as a function of AOM driving frequency, in the double pass configuration. The steps in the data are an artifact of data acquisition process. The AOM does produce some additional AM when the frequency control loop is engaged, shown in figure This is accounted for by the conversion of frequency modulations into intensity modulations, also shown on figure 3.32 is the frequency fluctuations of the AOM driving frequency multiplied by the FM to AM conversion factor. To calculate the effect of the additional AM generated by the AOM we multiply the additional AM by the AM to FM conversion factor. The expected frequency noise in the system from AM generated in the AOM is below 0.04 Hz/ Hz at 10 Hz and drops to Hz/ Hz at 1 khz. 62

79 3.7. ACOUSTO-OPTIC MODULATORS FOR FREQUENCY CONTROL (a) FM to AM conversion (b) AM to FM conversion Figure 3.31: (a) FM to AM conversion determined by modulating the AOM driving frequency by a known amount, measured at the reflected photodiode. (b) AM to FM conversion determined by introducing a known amount of AM at the AOM and measuring the FM in the frequency difference between the two cavities. Figure 3.32: Measured (red) and calculated (green) increase in RIN when the AOM is actively controlling the laser frequency compared to a constant frequency (blue) There is the possibility that the direction of the beam out of the AOM may fluctuate as the AOM driving frequency is changed to keep the laser locked, due to imperfect alignment of the cat s eye double pass configuration. The beam propagates through free space to reach the Fabry-Perot resonators and pointing fluctuations will cause changes in alignment into the cavity. It was calculated that without the cat s eye double pass configuration the beam 63

80 CHAPTER 3. FREQUENCY CONTROL OF A LASER would move 0.2 μm at the input mirror of the cavity for a 1 khz frequency shift. The cat s eye is designed to eliminate beam pointing changes but relies on the spacing between the lens and the mirror and the lens and the AOM being equal. It was estimated that the maximum error of the spacing is 1 cm which would result in only a 1% suppression of Bragg angle changes. For a 1 khz shift the expected beam motion would be 2 nm compared to a 188 μm mode size and from previous measurements of the cavities sensitivity to misalignment [Dawkins 08] this level of misalignment was calculated to cause a negligible frequency shift. 3.8 Conclusions Using the correction signal sent to the AOM presents a method of determining the frequency stability of the two optical resonators without the need for an optical beat note, greatly simplifying the experiment. However it was necessary to investigate the possible sources of error produced by the AOM, as the technique utilizes an inloop measurement that can give misleading results. The results of the experiments carried out in this section are shown in figure All measured and calculated frequency noise introduced by the AOM used for frequency control are well below 1 Hz/ Hz at the frequencies of interest, validating the technique. 64

81 3.8. CONCLUSIONS Figure 3.33: Summary of the in loop frequency noises introduced by the AOM when used for active frequency control - Synthesizer input noise (black), intrinsic measured AOM noise (green), FM noise from acoustic phase delay (red) and AM to FM conversion (blue) 65

82

83 It is not possible to fight beyond your strength, even if you strive Homer ca. 800 BC 4 Frequency Stability Chapter Overview This chapter concerns the sources of noise in the experiment that limit the frequency stabilization of the laser. The limitations are measured and attempts to reduce the levels of noise in the experiment are described. Additionally the stabilized laser was used to measure the frequency stability of an iodine frequency standard based on a hollow-core optical fibre. 4.1 Noise Sources in Optical Resonators There are limitations to the performance of a laser frequency stabilized with an optical cavity. These limitations can arise from noise present in the laser, the control system or the from an inherent property of the optical resonator. The dominant noise sources for the experiment are shown in figure 4.1 and this section will be a brief overview of their origins and properties. The subsequent sections detail the measurements of the limitations and attempts to improve the performance of the experiment. Cavity Vibration: Acceleration of the optical cavity will cause length fluctuations that will be converted into frequency fluctuations when the laser is locked to the cavity. Vibrations reach the optical cavity through the optical table and the cavity supports. The levels of vibration and length fluctuations are covered in section 4.2. Bandwidth Limit: If the control system has limited bandwidth there will be 67

84 CHAPTER 4. FREQUENCY STABILITY Figure 4.1: Block diagram of the dominant sources of noise in a laser that is frequency stabilized with an optical resonator. insufficient suppression of noise by the control loop due to limited gain. The bandwidth limit of the initial optical setup found in section 4.3 was overcome by the use of two AOMs for frequency control, as described in section Detection Noise: The Pound-Drever-Hall frequency locking technique requires demodulation of the reflected power at the modulation frequency. Detector noise around this frequency will create a noise floor below which frequency fluctuations cannot be detected and suppressed. The noise floor of the photodiode measuring the reflected signal can limit the measurement if it is above the intensity noise of the laser at the modulation frequency. Other noise floors may be present in amplifiers and mixers that perform the demodulation. The measurements of detection noise are presented in section 4.4. Intensity Noise: Similar to the detection noise, the Relative Intensity Noise (RIN) of the laser at the modulation frequency can dominate the detection of frequency fluctuations and create a noise floor. There is also Residual Amplitude Modulation (RAM) present in the laser at the modulation frequency. RAM is unwanted amplitude modulation of the laser caused by the PDH frequency modulation. The demodulation of RAM causes an offset in the error signal and variations in RAM can limit the frequency stability. Detector noise, RIN and RAM effects can be quantified by a measurement of the input to the 68

85 4.2. VIBRATION control loop V err when the laser is off-resonance, see section 4.4. Attempts were made to reduce the effects of intensity noise are described in section 4.6. Fundamental Noise - Laser Shot Noise and Cavity Thermal Noise: The quantum nature of light gives a fundamental limit on the amplitude noise of the laser. This will limit the frequency stability of the laser in the same way as RIN discussed above. There are also fundamental limits on the dimensional stability the cavities from the thermal fluctuations of Brownian motion and thermo-elastic damping. These are calculated in section Vibration The frequency stability of the two Fabry-Perot resonators, for the initial setup is showninfigure4.2 Figure 4.2: Relative frequency Stability of the two Fabry-Perot Cavities The spikes in the data from 10 Hz to 400 Hz are due to mechanical vibrations of the sapphire spacers. The heat shield surrounding the Fabry-Perot resonators is mounted on rubber O-rings to reduce vibration however, there will be some transmission of vibration to the resonators. To measure the motion of the spacer a simple Michelson style interferometer was created using the input mirror on the Fabry-Perot cavities as one arm when the laser frequency is well away from a cavity resonance, show in figure

86 CHAPTER 4. FREQUENCY STABILITY Figure 4.3: Simple Interferometer to measure the motion of the Fabry-Perot cavity As the length of the two arm changes there will be constructive and destructive interference, leading to a sinusoidal varying voltage at the photodiode. To perform a measurement of the motion of the Fabry-Perot cavity we tune the length of the interferometer arms so the output is half-way between constructive and destructive interference, in the linear region of the sinusoidal curve. In this condition small variations in arm length give a linear voltage change and we can convert this to a displacement measurement. The interferometer is sensitive to changes in length of either arm so the Fabry-Perot cavity was replaced with a standard mirror securely mounted to vacuum housing. This was done to obtain the noise floor of the measurement system and reveal motion purely associated with the optical cavities. Interferometers where built for both cavities and the results are shown in figure 4.4. There are peaks in the power spectral density of the displacement that correspond well the peaks in the frequency stability. (a) PSD of Displacement of OFR 1 (b) PSD of Displacement of OFR 2 Figure 4.4: PSD of displacement of the input mirror of the Fabry-Perot cavities, note the peaks at 25 Hz, 100 Hz and 300 Hz. The black traces are the noise floors of the interferometers. 70

87 4.2. VIBRATION To validate the interferometer measurement two simple models for cavity deformation were employed. Fluctuations in horizontal displacement are associated with acceleration and as the Fabry-Perot cavities undergo horizontal acceleration the cavity length will be altered by the stress-strain relationship [Roark 02]: ΔL L = ma EA 4.1 where m is the mass of the object, a is the acceleration, A is the cross sectional area and E is the Youngs modulus (for sapphire E = 400 GPa [Black 04]). The horizontal displacement may also be caused by variations in the tilt of the spacer which will cause the cavity to stretch from gravity as the tilt angle is changed [Chen 06b]. In this case: ΔL L = ρl 2E g sin(cos 1 (1 Δx L )) 4.2 where ρ is the density of sapphire, g is acceleration due to gravity and Δx is the displacement at the end of the spacer. The measured displacement was entered into both these models and the expected frequency fluctuations are shown in figure 4.5 for OFR 1. Although the interferometer experiment has a poor signal to noise ratio and it is not clear whether horizontal or vertical acceleration is causing the unwanted frequency fluctuations, the conclusion can still be made that the peaks seen at low frequency are due to mechanical vibration of the Fabry-Perot cavities. It would be pointless to strive for better frequency control if mechanical vibrations limited the frequency stability of the cavities at higher frequencies. To measure the susceptibility of the experiment to vibrations in the environment we subjected the Fabry-Perot cavities to enhanced levels of vibration from a stereo loudspeaker. The speaker was placed on the optical table and a function generator was used to provide a sine wave at a particular frequency. The amplitude of the induced vibrations were measured with a Sci-Flex SF1500SA accelerometer on the optical table. The amount of vibration was far above the background level and caused a spike in the frequency stability between the two cavities which was recorded, along with the amount of vibration from the accelerometer. Figure 4.6 shows the size of the peak as the frequency is changed, normalized for a fixed amount of acceleration, also on the graph is the frequency stability of the Fabry-Perot cavities for comparison. The 71

88 CHAPTER 4. FREQUENCY STABILITY Figure 4.5: Frequency fluctuations from the motion of the Fabry-Perot cavities inferred from the measured displacement of the cavities (see figure 4.4). The green is trace is assuming tilt (equation 4.2) and the red trace is assuming displacement (equation 4.1). The blue trace is the frequency stability and the black traces are the noise floors of the detection systems. vibration sensitivity of the the cavities drops off significantly above 400 Hz, suggesting that we will not be limited by vibrational noise in the Fabry-Perot cavities at higher frequencies. 4.3 Bandwidth Limits The limit to the frequency stability of the experiment at higher frequencies is the low bandwidth of the PZT actuator used for frequency control of the laser. As shown in Section 3.2 the reduction of output noise is equal to 1 1+G(s)P (s) 4.3 where G(s)P(s) is the open loop gain of the system. Because of the resonance in the PZT at 10 khz we used a PI controller with a break point at 10 khz and the gain must be well below unity at this frequency. If the output noise of the laser is divided by the residual noise we get the open loop gain of the system (see figure 4.7) and this reaches unity at 2 khz, the bandwidth of the PZT control loop. At lower frequencies the integrator of the PI controller dominates and gain increases 72

89 4.3. BANDWIDTH LIMITS Figure 4.6: Susceptibility of the experiment to vibration (red data points) measured by introducing vibration through a stereo loudspeaker. The frequency stability is also shown (blue trace) scaled to match the vibration susceptibility, as an illustration of the expected contribution to instability from vibration above 400 Hz. Figure 4.7: Evaluation of the unity gain point of the PZT (OFR 1 ) control loop, inferred from the suppression of frequency noise with the loop engaged. The unity gain point is 2 khz to avoid oscillations. The dips from Hz are the control loop reaching the noise floor of the system, namely the mechanical vibrations of the two Fabry-Perot cavities. 73

90 CHAPTER 4. FREQUENCY STABILITY as 1/f, meaning we can expect the noise to be suppressed by a factor of 10 at 200 Hz - this is not enough gain to reduce the free-running fluctuations in the laser frequency below 3 Hz. Increasing the gain will only cause the system to oscillate, note the noise amplification already occurring at 10 khz in figure Adding a second integrator to the PZT based control loop (OFR 1 ) would boost the low frequency gain and this was implemented. Unfortunately to avoid oscillations the cut-off frequency of the second integrator must be so low that the increased gain above 400 Hz is negligible and there was no improvement to the noise floor. The mechanical vibrations below 400 Hz limit the frequency stability of the experiment and the increase in gain has no effect, so the second integrator was abandoned. To improve the frequency stability of the experiment there are a number of ways forward. The mechanical vibrations could be reduced by better mechanical isolation or active stability control. However the level of noise introduced by vibration is well characterized and advanced mechanical stability of Fabry-Perot cavities was not a priority of the experiment, so no further improvements were made on this front. The bandwidth limit imposed by the PZT control of the laser does hinder progress towards the noise floor of the Fabry-Perot cavities so it was decided to supplement the PZT frequency control of the laser with an AOM for fast frequency control Dual AOM setup As shown previously (section 3.7) an Acousto-Optic Modulator (AOM) enables frequency control much faster than the piezo-electric (PZT) transducer on the laser crystal; thus a double-pass AOM in the cat s eye configuration was added to the frequency control loop on OFR 1 to increase the control bandwidth. The optical setup remains essentially the same except the light frequency locked to OFR 2 is now separated after the AOM so the signal sent to the AOM still represents the frequency difference between the two Fabry-Perot cavities. The optical layout for the improved setup is shown in figure 4.8. A schematic view of the frequency control system for OFR 1 is shown in figure 4.9. The AOM, signal generator and amplifier are identical to those used previously, 74

91 4.3. BANDWIDTH LIMITS Figure 4.8: Optical setup with two AOMs. AOM 1 provides high frequency frequency corrections to supplement the bandwidth of the PZT on the laser used to lock the laser to OFR 1 a Crystal Technologies , Agilent N181A and mini-circuits ZHL-1-2W respectively. Again the FM input of the signal generator is used for frequency control of the AOM (and hence the frequency of the laser light), and the filtering and demodulation of the error signal is essentially the same. The PDH modulation is still provided by the PZT on the laser crystal as before. Figure 4.9: Frequency control setup for OFR 1 using an AOM for fast frequency control. The integrator at the output of the Lock Box used for the PZT control loop has gain designed to dominate the AOM 1 control loop at frequencies less than 100 Hz. The reduction in output sent to the AOM (see figure 4.10) is evident as the amount 75

92 CHAPTER 4. FREQUENCY STABILITY of frequency control done by the AOM drops below 100 Hz, this ensures the AOM only contributes to the fast frequency control of the laser, which requires only small frequency changes and it remains at its optimal operation frequency of 80 MHz. Figure 4.10: Frequency correction sent to AOM 1, the drop below 100 Hz is due to the PZT control loop dominating control of the laser. In addition the AOM for OFR 1, frequency control new photodiodes were installed in the system. The old New Focus 2031 Si photodiodes were replaced with Thorlabs PDA10CS InGaAs detectors. These photodiodes have a higher responsivity to 1064 nm light and much higher bandwidth due to the reduced detector size. The PDH modulation frequency was also increased from khz to khz to take advantage of the increase in detector speed and the modulation depth and filtering of the error signal was adjusted accordingly. The result of the improvement to the laser frequency stabilization setup is shown in figure The increase in bandwidth of the OFR 1 frequency lock has enabled us to increase the gain of the control loop and suppress frequency fluctuations of the laser to a much higher extent. 4.4 Detection and Intensity Noise To understand the limiting factor in the improved setup it is necessary to return to the Pound-Drever-Hall locking scheme. The frequency discriminator provided by the PDH technique requires the demodulation of the error signal from the modulation 76

93 4.4. DETECTION AND INTENSITY NOISE Figure 4.11: Frequency stability of the initial (blue) and dual AOM (red) optical resonator experiments, measured using the AOM correction (see section 3.6). The higher bandwidth of the AOM enables higher gain in the frequency region of interest (400 Hz - 10 khz) frequency to DC. Noise in the detection electronics and laser around the modulation frequency will also be demodulated to DC by the mixer and will represent a noise floor in the frequency locking system. Noise in Volts at the input to the control system can converted to frequency noise using the measured slope of the frequency discriminator. In the detection system there is noise present in the demodulation system (mixers and amplifiers, see figure 4.9) and the photodiode. The noise floor of the demodulation electronics can be ascertained by measuring the noise of the system with no photodiode and the noise in the photodiode can be measured when there is no laser light present. It was found that the noise in the photodiode detector dominated the noise in the demodulation electronics the results for OFR 1 are shown in figure 4.12 with all noise floors are below 1 Hz/ Hz by 1 khz for both systems. Also included on the graph is noise floor using the old photodiodes, and the much higher noise was the major motivation for switching to the new photodiodes. The intensity noise of the laser at the modulation frequency can be tested by measuring the noise floor of the detection system when the laser is well away from a resonance of the Fabry-Perot cavity. In this situation there is no frequency dis- 77

94 CHAPTER 4. FREQUENCY STABILITY Figure 4.12: Noise floors from the demodulations system and photodiodes determined by measuring the Voltage noise at the input to the frequency control system and converting to frequency noise. The blue trace is the previous photodiodes (New Focus 2031) and the green and black traces are the new photodiodes (Thorlabs PDA10CS). criminator and the noise at the input of the control loop is the intensity noise of the laser de-modulated to DC, once the drop in reflected laser intensity on resonance has been taken into account. An off-resonance measurement of laser intensity noise at the modulation frequency encompasses both Relative Intensity Noise (RIN) and Residual Amplitude Modulation) RAM of the laser, though the source of each noise type is different. RIN is the intensity noise of the laser, which will ultimately be limited by the the shot noise of the laser. RAM is unwanted amplitude modulation caused by the frequency modulation required for the PDH locking scheme. Tests were carried out for both OFR 1 and OFR 2 control loops and the offresonance noise for both Fabry-Perot cavities was measured. Also contributing to the limit of frequency stability is the residual error of the control system - the noise floor of the control loop. In figure 4.13 the off resonance noise of both cavities and the residual errors are combined to create the total noise floor and compared to the measured frequency stability of the two Fabry-Perot cavities. The laser intensity noise is above the detector noise (figure 4.12) and creates the white noise floor seen above 400 Hz, which unlike the photodiode noise scales with laser intensity 78

95 4.5. FUNDAMENTAL LIMITS and cannot be mitigated by increasing the amount of light. From this figure we can conclude that the stability of the Fabry-Perot optical frequency standards are limited by intensity noise in the laser at the PDH modulation frequency. An attempt to reduce this technical limit will follow but first an overview of the fundamental limits of the cavities will be presented. Figure 4.13: Total noise floor of the frequency stabilization system (blue) compared to the frequency stability of the two Fabry-Perot cavities (red). 4.5 Fundamental Limits There are fundamental limits to the stability of an optical resonator. These limits can be due to dimensional uncertainty in the distance between the two mirrors or the quantum mechanical nature of the light. While the optical system has been completely revised, the two Fabry-Perot cavities used by S. Dawkins remain identical [Dawkins 07] and the same fundamental limits apply. In this section the intrinsic length fluctuations of the Fabry-Perot cavities are expressed in the frequency domain as G(f) and converted to frequency noise S(f) through equation Brownian Motion Brownian motion in the spacer, mirror coatings and mirror substrates all contribute to fluctuations in the distance between the two mirrors that define the resonant 79

96 CHAPTER 4. FREQUENCY STABILITY frequencies of the Fabry-Perot cavities. The Brownian motion contributing to the frequency uncertainty of the optical cavities is determined from [Numata 04], which uses a structural damping model. For a spacer of length L and radius R we have: G spacer (f) 4k BT 2πf L 3πER 2 φ spacer 4.4 where k B is Boltzmann s constant, T is the average temperature, E is the Young s modulus and φ spacer is the mechanical loss angle of the spacer material, which is equivalent to the reciprocal of the mechanical quality factor. The contribution from the mirror substrate is: G sub (f) 4k BT 2πf 1 σ 2 πeω0 φ sub 4.5 where ω 0 is the beam spot size on the mirror, σ and φ sub are the Poisson s ratio and the angle of loss of the substrate material respectively. The Brownian motion in the mirror coating is related to the fluctuations in the substrate: ( G coat (f) G sub (f) 1+ 2 ) 1 2σ φ coat d π 1 σ φ sub ω where φ coat is the coating material angle of loss and d is the coating thickness. From equation 4.4 the advantage of using sapphire should be apparent, Brownian noise is inversely proportional to the Young s modulus and also lowered by the mechanical loss of the spacer material. Sapphire has a larger Young s modulus then fused silica or ULE (Ultra-Low Expansion) glass, but more significantly, the mechanical loss of sapphire is two orders of magnitude lower. Combining the Brownian length fluctuations of the spacer, mirror substrates and coatings (with the assumption that the noise sources are uncorrelated) we get a total frequency noise of 0.08 Hz/ Hz at 1 Hz for OFR 1. The fused silica mirror substrate and mirror coatings contribute to 99% of this value and the remainder is from the sapphire spacer. OFR 2 has frequency noise of 0.1 Hz/ Hz at 1 Hz, which is dominated by the mirror coatings (contributing 98 %) as the rest of cavity is constructed from sapphire. Hence the expected frequency stability limit caused by Brownian motion in the cavity is well below the 1 Hz/ Hz level at 1 Hz and the inverse relationship with frequency further reduces the limit at higher frequencies. 80

97 4.5. FUNDAMENTAL LIMITS Thermo-Elastic Damping There is another source of dimensional fluctuations because of the non-zero temperature of the optical cavities. This noise source is independent from Brownian motion and would be absent in a cavity made from ULE. Thermoelastic damping is associated with thermodynamic fluctuations in temperature about some mean value and the resulting fluctuations in the volume of materials with a non-zero coefficient of thermal expansion. We use the model of [Cerdonio 01] which is valid below the adiabatic limit, when the thermal time constant is equal to the beam spot size: G sub = 8 α 2 (1 + σ) 2 k BT 2 ω 0 J( f ) 2π 2k f 4.7 c where α is the coefficient of thermal expansion, k is the thermal conductivity and the adiabatic limit is given by f c = k 2πCρω 2 0 with ρ the density of the material and C the specific heat. The shaping function J( f f c ), which reduces the effect at frequencies below the adiabatic limit, is given by: J( f 2 u 3 e u2 /2 )= du dv f c π 0 (u 2 + v 2 )[(u 2 + v 2 ) 2 +(f/f c ) 2 ] 4.8 The adiabatic limit for the the fused silica mirrors of OFR 1 is f c 4 Hz and for the sapphire mirrors of OFR 2 is f c 180Hz. The expected noise from thermoelastic damping due to the mirrors is 0.03 Hz/ Hz and 0.1 Hz/ Hz at 1 Hz for OFR 1 and OFR 2 respectively. To evaluate the contribution of thermoelastic damping from the mirror coating we use the model of [Braginsky 03] which is valid as the beam spot size is significantly smaller than the dimensions of the mirror: G coat (f) 8 α 2 (1 + σ) 2 2k BT 2 d 2 1 2π kρc 2πf ω The mirror coating of OFR 1 has thermoelastic noise at 1 Hz of 0.01 Hz/ Hz and OFR 2 at 0.05 Hz/ Hz. As the spacer material used in these experiments (sapphire) has a non-zero coefficient of thermal expansion we must consider the thermoelastic noise in the spacer. The model used here is found in [Dawkins 07] which was based on the approach of [Braginsky 03]. Analysis of thermodynamic temperature fluctuations in the spacer yields: G spacer (f) k BT 2 α 2 L k

98 CHAPTER 4. FREQUENCY STABILITY using a semi-quantative approach. A determination of thermo-elastic damping from analysis of phase fluctuations in an optical fibre [Wanser 92], appropriately modified to apply to a Fabry-Perot cavity gives a results differing by 1/(8π 2 ) [Dawkins 07]. From the formula above the result is a white (frequency independent) noise floor at 0.6 Hz/ Hz or 0.06 Hz/ Hz if the [Wanser 92] result is used Laser Shot Noise The quantum nature of light creates another fundamental limit for the frequency stability of an optical resonator. As stated in the previous section noise in the detection system at the PDH modulation frequency is indistinguishable from the error signal used to frequency lock the laser. independent and is given by [Day 92]: S shot (f) = ν cav J 0 (β) The shot noise limit is frequency hf 8ηP det C 4.11 where h is Heisenberg s constant and f is the frequency of the laser. The optical power falling on the photodiode is P det and η is the quantum efficiency of the photodiode. The cavity bandwidth is ν cav and C is the contrast of the optical cavity. The optical power used in these experiments was 100 μw and the PDA10CS photodiodes have a quantum efficiency of 0.8, which gives a shot noise limit on the frequency stability of the cavities of 0.02 Hz/ Hz at all frequencies. 4.6 AOM for Fast Laser Intensity Control Evidently the fundamental limits to the performance of the optical cavities are below the technical noise limits caused by intensity noise of the laser at the modulation frequency. To reduce the noise of the laser at this frequency the AOMs used for frequency control can also be employed for intensity control. The amount of light diffracted, and hence frequency shifted and output by the AOM, is proportional to the RF power sent to the device [Young 81]. By varying the RF power reaching the AOM the laser intensity can be controlled. The experimental setup was slightly modified to bring the transmitted laser beam position as close as possible to the 82

99 4.6. AOM FOR FAST LASER INTENSITY CONTROL piezo-electric actuator to increase the response speed of the AOM, furthermore an independent photodiode was employed before the optical cavities to measure the intensity noise, as using the reflected photodiodes would interfere with the frequency lock by eliminating the PDH signal through the intensity control. As the laser out of AOM 1 is common to both cavities only AOM 1 needs to be used for intensity control. The AM input of the signal generator did not have the bandwidth required ( 400 khz) so a Voltage Variable Attenuator (VVA) was used to vary the power reaching the AOM. An M/A-Com AT-259 VVA was placed between the signal generator and the RF amplifier; with a 5 ns rise time this device is more than fast enough for intensity control at 404 khz. The level of attenuation provided by the VVA is controlled simply through the control voltage (1V to -3V) and the attenuation ranged from -3 db to -8dB for the 80 MHz RF that was sent to the AOM. The response of the VVA is non-linear but we are rectifying intensity noise at a very small level so the action of the VVA can be linearized around the operation point. The scheme is shown in in figure 4.14 and the transfer function of the combined VVA and AOM laser intensity control system was measured, and the results are shown in figure We believe the the rapid cut-off around 1 MHz is limited by the acoustic wave transit time in the AOM. Since only intensity noise in the laser near (± 50 khz) the PDH modulation frequency (404 khz) needs to be reduced we can use a control loop deviating from the traditional PID approach. The frequency at which we wish to control the intensity of the laser is close to bandwidth of the AOM so it would be difficult to generate enough gain in the control loop using a PI controller, thus a Sallen-Key band-pass filter was used [Chen 86]. A Sallen-Key (SK) topology uses a single op-amp to create a band-pass filter with a Q-factor and the electronics were designed to have the resonant frequency at the PDH frequency. This enables more gain at a specific frequency than using an integrator while ensuring the stability of the control loop. The measured transfer function of the Sallen-Key band-pass filter is shown in figure The aim of using a VVA and AOM for fast laser intensity control was to reduce the amount of AM noise at the PDH modulation frequency of khz. However the system was not stable at this frequency because it was too close to the cut-off 83

100 CHAPTER 4. FREQUENCY STABILITY Figure 4.14: The power of the RF signal sent to AOM 1 is varied by the VVA to actuate changes in the laser intensity reaching the Fabry-Perot cavities to reduce the RIN of the laser at the PDH modulation frequency. Figure 4.15: VVA and AOM transfer function, measured through laser intensity at the control photodiode frequency of the AOM. It was decided to reduce the PHD modulation frequency to khz, where the phase delay from the SK band-pass filter combined with the phase delay of the VVA and AOM (figure 4.15) is still small enough to keep the control loop stable. The reduction of RIN at the reflection photodiode for OFR 1 isshowninfigure 84

101 4.6. AOM FOR FAST LASER INTENSITY CONTROL Figure 4.16: Transfer function of the Sallen-Key bandpass filter, the resonant frequency was chosen as 200 khz and the Q-factor is , it was measured two ways, by the photodiode in the control loop and monitor photodiode outside the control loop. There is significant reduction of the RIN at khz, showing the validity of our approach. As expected there is no reduction of noise at other frequencies because the control loop has small gain at other frequencies. The amplification of noise at 530 khz illustrates how close to the oscillation threshold the control loop was, and emphasizes the benefit of using a filter with Q to increase the gain in at a specific frequency, as opposed to an integrator. There is a factor of 10 decrease of RIN in the in-loop measurement at the PDH frequency, however the out of loop measurement has only a factor of 3 decrease, this is due to the noise floor of the in-loop photodiode, a New Focus 2031 (see figure 4.12). The intensity control loop reduced the RIN of the laser at khz to the level of RIN at khz, the previously used PDH modulation frequency. The approach has shown that is is possible to control the power of a laser through an AOM and VVA to a high degree and validated the approach of a band-pass filter with Q in a control loop, although the efforts provided no improvements in the frequency stability of the experiment. It may be possible to increase the bandwidth of the AOM through a smaller beam size closer to the piezo-electric transducer on the AOM. Given the limitations of the AOM control it would be possible to implement much faster control with an electro-optic modulator but this was beyond the scope 85

102 CHAPTER 4. FREQUENCY STABILITY (a) In-Loop (b) Out of Loop Figure 4.17: In-loop and out of loop reduction of laser RIN by a VVA and AOM using a band-pass filter with Q, the control loop is locked (green) and unlocked (red). The PDH modulation frequency is khz. of the project. However due to changes of focus in the project such endeavors where not undertaken and the Sallen-Key scheme to reduce intensity noise at the PDH modulation frequency was concluded after a proof of concept. To validate the performance of the frequency standard one of the Fabry-Perot cavities (OFR 1 ) was to be used as frequency reference to measure the stability of another project in the group, namely an Iodine fibre frequency standard (see section 4.7). This measurement was concerned with frequency stability on the timescales of 1 second or greater, so the AOM based intensity servo was easily re-configured to reflect this change in priority. The transmitted power through the Fabry-Perot cavity was used in conjunction with an integrator control loop to stabilize the power circulating in the optical cavity (as shown in figure 4.18), minimizing the influence of optical power variations on the stability of the cavities through optical-thermal effects. The reduction of intensity fluctuations is shown as a SRAV in figure 4.19 with the calculated fractional frequency fluctuations such variations in the intensity would cause. The amount of frequency instability caused by the intensity fluctuations are derived from a previous measurement of Hz/W [Dawkins 07] and such fluctuations are below the expected frequency stability of the iodine fibre reference. 86

103 4.7. USE AS A FREQUENCY REFERENCE Figure 4.18: Transmission intensity control loop. The optical power transmitted circulating in OFR 1 is held constant by the control loop Figure 4.19: SRAV of transmitted power with control loop locked (red) and unlocked (blue). The fractional frequency fluctuations are inferred from previous measurements of the response of OFR 1 to power fluctuations 4.7 Use as a Frequency Reference One of the Fabry-Perot cavities from this project was used as an optical frequency reference to measure the frequency stability of an Iodine fibre frequency reference. An iodine frequency reference uses a laser locked to a transition in iodine gas to provide a primary frequency reference. In this experiment the iodine was loaded into several meters of Hollow Core Photonic Crystal Fibre (HC-PCF) instead of the 87

104 CHAPTER 4. FREQUENCY STABILITY conventional large gas cell. The use of HC-PCF presents advantages in compactness and potential performance and it was necessary to verify the frequency stability of the iodine reference by creating a beat note with a stable reference laser. The atomic transition used to lock the laser to the iodine is at 532 nm but the laser used for the iodine HC-PCF experiment is a frequency doubled Nd:YAG laser with an output port for 1064 nm which can be used to create a beat note with the stable Fabry-Perot cavity used in this project. The schematic for the frequency stability comparison is shown in figure Figure 4.20: Measurement of frequency stability of HC-PCF iodine frequency reference. The HC-PCF experiment was located in another room and the light was transported to the Fabry-Perot optical frequency reference through 10m of single mode optical fibre. An optical beat note was generated and measured by a frequency counter. Unlike the previous work that focussed on fast frequency fluctuations in an attempt to reach a new noise floor, this work required frequency stability on timescales of 1 second and greater. To verify the frequency stability of the cavity stabilized lasers initially the two independent cavity stabilized lasers were compared to create beat note at 160 MHz which was measured by an EOT-3000A 2GHz amplified InGaAs photodetector. To measure the frequency stability of the Iodine HC-PCF the laser was locked to the fundamental mode of OFR 1 nearest to the iodine transition, which in this case created a 705 MHz beat note. The beat note was amplified, 88

105 4.7. USE AS A FREQUENCY REFERENCE filtered and recorded by a frequency counter and the results of the experiment are shown in figure Figure 4.21: SRAV of the frequency stability of the laser used in the iodine HC- CPF experiment unlocked (red), locked to an iodine cell (blue) and locked to an iodine HC-CPF fibre (black). The bottom line (purple) is the frequency stability of the two Fabry-Perot cavities (with linear drift removed), indicating they are sufficiently stable for these measurements. The frequency stability measurements have the long term temperature driven frequency drift of the cavities removed. The temperature of the sapphire spacers was recorded concurrently with the beat note and a linear fit was used to removed the drift, with the frequency drift rate consistent with the measured temperature drift rate and the known frequency/temperature dependence. While the performance of the frequency reference OFR 1 at 1second is not at the limits of optical cavity frequency stabilization the system is more than adequate for this purpose with frequency fluctuations at 1 second an order of magnitude below the iodine HC-PCF frequency standard. Using OFR 1 as a reference laser we were able to show a ten-fold improvement in frequency stability for gas-loaded HC-PCF frequency standards at 1 second [Lurie 11]. 89

106 CHAPTER 4. FREQUENCY STABILITY 4.8 Conclusions The implementation of two sapphire Fabry-Perot resonators have yielded several key results: We have shown that AOMs can be used for fast frequency control with minimal spurious effects and high bandwidth. The use of AOMs in a dual optical resonator system negates the requirement for an optical beat note which simplifies the experimental process enormously, and, using this system we have reached a technical limit of the experiment. Using an AOM for frequency control also has the supplementary benefit of intensity control which was combined with a novel control loop in an attempt to reduce the laser intensity noise at the PDH modulation frequency. The limit reached by the experiment was a fractional frequency stability of 1.7 Hz/ Hz ( / Hz) at 1 khz limited by the RIN noise of the laser and only an order of magnitude above the fundamental limits. While the frequency stability on longer time scales is limited by vibration and temperature drift, the system was successfully used as a frequency reference. Figure 4.22: Fractional frequency stability of the two Fabry-Perot optical cavities. The fractional frequency stability (red) white noise floor at / Hz is limited by the RIN of the laser (blue). 90

107 The only way of finding the limits of the possible is by going beyond them into the impossible Arthur C. Clarke ( ) 5 Tests of Lorentz Invariance Chapter Overview As one of the foundations of modern physics the concept of Lorentz invariance has been the subject of increasingly precise experimental verification. The Standard Model Extension (SME) provides a framework in which tests of Lorentz invariance can be compared and a description of the theory is provided. Focussing on the photon sector of the SME, the idea of an odd-parity resonator is introduced and the current constraints on the parameters of the standard model extension are reviewed. 5.1 Background Over the last 120 years there have been many tests of Lorentz Invariance (LI), although the motivation and apparatus have evolved significantly from the first famous experiments. The original tests of LI were performed by Michelson and Morley [Michelson 87], a series of experiments designed to measure the velocity of the earth with respect to the luminiferous aether, the accepted medium of light propagation at the time. The experiment was based on the Michelson interferometer, where a beam of light is split into two orthogonal arms and recombined after reflection from mirrors. The apparatus is sensitive to the relative phase shift experienced by the two paths of the light propagating down and back the two arms of the apparatus, appearing as interference fringes at the output. The motion of the earth relative to the luminiferous aether was expected to produce a relative phase shift in the interferometer arm parallel to the motion of the earth. By rotation of the experiment in 91

108 CHAPTER 5. TESTS OF LORENTZ INVARIANCE the laboratory the orientation of the two arms of the experiment are continuously switched and the fringes were expected to exhibit periodic variation. These series of experiments culminated with the famous null result - the motion of the earth could not be measured above the experimental noise floor. (a) (b) Figure 5.1: taken from [Michelson 87]. Perspective (a) and plan (b) of the Michelson Morley experiment, Interferometer experiments continued to improve and gather null results [Morley 05, Kennedy 26, Joos 30], becoming instrumental in the dismissal of the luminiferous aether and the widespread acceptance of Einstein s special relativity, published in 1905 [Einstein 05]. Special relativity elegantly reconciled the conflicts between contemporary theory and experiment, providing a framework on which modern physics has been built. The two principle postulates of special relativity are: 1. The principle of relativity 2. The constancy of the speed of light The concept of Lorentz invariance is a consequence of these two postulates and requires the laws of physics to be the same in all inertial reference frames. At the laboratory level the concept of local LI dictates that the results of an experiment are independent of the orientation or velocity of the experiment. The prominence of special relativity led away from the proof by contradiction of the aether experiments towards verifying a kinematic framework consistent 92

109 5.2. THE STANDARD MODEL EXTENSION with special relativity. The Robertson-Mansouri-Sexl (RMS) framework was widely used, first developed in 1929 [Robertson 49] and extended in 1977 [Mansouri 77a, Mansouri 77b]: it assumes the existence of a preferred reference frame and allows for violations of LI. By parameterization of the LI violations, various experimental results can be directly compared to the expected values consistent with special relativity, which is assumed to be correct. Deviations from the expected values represent a breakdown of LI. Concurrently with the new attitude towards LI the technology moved from interferometers to frequency standards (usually resonant cavities) where the observable becomes the frequency of the electromagnetic wave instead of phase shifts. A significant test in this era used a single microwave resonator on a continuously rotating table referenced to a quartz oscillator [Essen 55]. Another interesting test was performed using counter-propagating ammonia beam masers, where the frequency difference of the masers was recorded, the apparatus rotated 180 and the difference frequency recorded again [Cedarholm 58]. A similar experiment was performed using two orthogonal He-Ne lasers [Jaseja 64] where the frequency difference was monitored as the experiment oscillated back and forth through 90. A major increase in sensitivity came with the use of a rotating He-Ne laser referenced to a stationary methane laser [Brillet 79]. The reduction of systematic effects and better data acquisition and analysis techniques led to an increase in sensitivity over previous experiments by a factor of However all experiments undertaken reported no deviation from special relativity and a period of experimental inactivity followed, which was broken in 2003 with a renewed interest in tests of Lorentz Invariance that has continued to this day. 5.2 The Standard Model Extension Modern tests of Lorentz invariance use the Standard Model Extension (SME) which provides a general framework for the analysis of experimental tests of LI [Colladay 98, Colladay 97]. The development of the SME was driven by theoretical developments where the discussion moved beyond special relativity towards the unification of 93

110 CHAPTER 5. TESTS OF LORENTZ INVARIANCE the standard model with general relativity. Some unified theories predict a breakdown of Lorentz Invariance at some level ([Kostelecký 89, Kostelecký 91, Alfaro 02, Bjorken 03]) which motivated a more comprehensive framework. The SME contains all possible Lorentz and CPT violations by known fields. As a comprehensive theory there are particle, gravity and photon sectors of the SME, however the work reported in this thesis is concerned with the propagation of light and we restrict ourselves to the photon sector. In the photon sector of the SME the standard Lagrangian: L = 1 4 F μμf μν 5.1 is extended to include all possible violations of Lorentz invariance by known fields to become [Kostelecký 02]: L = 1 4 F μνf μν 1 4 (k F ) κλμν F κλ F μν (k AF ) κ ɛ κλμν A λ F μν 5.2 where A λ is the electrodynamic 4-potential and F μν is the electrodynamic field tensor. The tensor (k AF ) κ has the dimension of mass and are CPT odd terms. In the minimal SME these terms are set to zero as they introduce instabilities in the Langragian leading to non-renormalizability. It should be noted that the SME has been developed to include non-renormalizable terms of higher mass dimension [Kostelecký 09] and the analysis has recently been carried out for certain resonant cavity experiments [Parker 11]. The effect of higher-order Lorentz violations are not considered here and the analysis is restricted to the minimal SME, though analysis for higher dimensional parameters may be undertaken in the future. The remaining (k F ) term is CPT even, dimensionless and has 19 independent Lorentz violating parameters. Applying the standard procedure of minimizing the action for equation 5.2 using variational techniques and the definitions F μν μ A ν ν A μ and A μ (φ, A) we obtain the equations of motion of the system. The equations for the photon sector of the minimal SME are analogous to Maxwell s equation for the propagation of light in an anisotropic medium. The 94

111 5.2. THE STANDARD MODEL EXTENSION parameters of k F are arranged in linear combinations: (κ DE ) jk = 2(k F ) 0j0k (κ HB ) jk = 1 2 ɛjpq ɛ krs (k F ) pqrs (κ DB ) jk = (k F ) 0jpq ɛ kpq (κ HE ) jk = (κ DB ) kj and the resulting dynamics can be described in terms of modified D, H and E, B relations and the standard Maxwell equations apply for the propagation of light. D ɛ 0 ( ɛ r + κ DE ) ɛ0 μ 0 κ DB = E H ɛ0 μ 0 κ HE μ 1 0 ( μ r + κ HB ) B Thus, the minimal SME in the photon sector introduces the properties of an isotropic medium to the propagation of light in the vacuum with the standard relations recovered in the absence of Lorentz invariance (κ DE = κ HB = κ DB = κ HE = 0). Non zero values of the κ matrices are considered to be remnants of Planck scale physics in the early universe and if one or more parameters are zero it has no implications for the values of the other κ matrices, they are independent of each other. Further linear rearrangement of the parameters to four 3 3 matrices and a scalar generate forms that are accessible to experiments. ( κ e+ ) jk = 1 2 (κ DE + κ HB ) jk ( κ e ) jk = ( 1 2 (κ DE κ HB ) jk 1 3 δjk (κ DE ) ll ( κ o+ ) jk = 1 2 (κ DB + κ HE ) jk ( κ o ) jk = 1 2 (κ DB κ HE ) jk κ tr = 1 3 (κ DE) ll 5.4 In the above equation the subscript e refers to even-parity matrices, the subscript o refers to odd-parity matrices and the parameter κ tr is a scalar. Different types of experiments are sensitive to the various parameters depending on the physical manifestation of the Lorentz violating effect. 95

112 CHAPTER 5. TESTS OF LORENTZ INVARIANCE Broadly speaking the the parameters of the SME can be assigned the following properties: The κ e+ and κ o represent birefringence of the vacuum. The remaining SME parameters together create a time-independent but orientation-dependent modification to the speed of light parameterized in the Sun-centered Celestial Equatorial reference frame (SCCEF). Essentially the isotropic κ tr is an average shift over all possible directions, κ jk e represents a directional dependence in the speed of light and κ jk o+ is the relative difference between light moving parallel and anti-parallel to some particular direction. Laboratory based experiments are sensitive to combinations of these SME parameters through Lorentz transforms from the SCCEF to the laboratory reference frame and a time dependence on the parameters is induced through the relative orientation of the earth. For terrestrial experiments the properties and detection techniques used for the SME parameters are summarized in the table below. Table 5.1: The photon sector parameters of SME Parameter Physical Manifestation Detection Technique κ e+, κ o Birefringence Astrophysical polarimetry measurements κ e Anisotropic shift Even-parity (Michelson-Morely type) resonator tests κ o+ Counter-propagating relative shift Even-parity (Michelson-Morely type) resonator tests κ tr Isotropic shift Odd-parity Resonator tests, Ives- Stilwell experiments, particle physics The κ e+ and κ o parameters are birefringent terms and are constrained to a very small level by astrophysical observations [Kostelecký 02]. The modified dispersion relation caused by birefringent LI violations gives rise to two distinct perpendicular polarization modes with different phase velocity. Distant pulsed light sources such 96

113 5.3. CHOICE OF REFERENCE FRAME as pulsars or gamma ray bursts would be split into two polarization modes with an arrival time delay created by the distance traversed by the pulse. In the presence of birefringent LI violations broadband sources of polarized light will encounter a relative phase shift dependent on wavelength of the light that will induce changes in polarization over the emitted spectrum. The analysis of astrophysical data [Kostelecký 02] has limited the magnitude of possible birefringent LI violations to < As this constraint is many orders of magnitude lower than is expected from terrestrial experiments κ e+ and κ o are set to zero when they appear in the analysis of cavity based experiments. The nonbirefringent κ e, κ o+ and κ tr parameters are constrained by terrestrial experiments and for this we require a reference frame in which the parameters are assumed to be constant. 5.3 Choice of Reference Frame In the standard model extension violations of Lorentz invariance are assumed to be remnants of Planck scale structure in the early universe [Kostelecký 02]. Therefore the Cosmic Microwave Background is chosen as the reference frame in which the SME parameters are considered constant. A Sun-centered Celestial Equatorial reference frame (SCCEF) is sufficiently inertial to the CMB to provide the basis for analysis of Lorentz invariance experiments and Lorentz transformations between the SCCEF and the laboratory give rise to the modulation of the SME parameters. The SCCEF has the Z axis pointing towards the celestial north pole, the X axis points from the sun towards the earth at the moment of the autumnal equinox, and the Y axis is chosen in accordance with the right hand rule, see figure 5.2. In the laboratory reference frame the local coordinates are defined on the surface of the earth. The z axis is defined normal to the ground pointing vertically upward, the x axis points north and the y axis points east. The Lorentz transformation from the SCCEF to the stationary laboratory frame 97

114 CHAPTER 5. TESTS OF LORENTZ INVARIANCE Figure 5.2: Sun-centered Celestial Equatorial reference frame (SCCEF) used in the SME. The SCCEF has the Z axis pointing towards the celestial north pole, the X axis points from the sun towards the earth at the moment of the autumnal equinox, and the Y axis is chosen in accordance with the right hand rule. is given by: Λ μ ν = 1 β 1 β 2 β 3 (R β) 1 R 11 R 12 R 13 (R β) 2 R 21 R 22 R 23 (R β) 3 R 31 R 32 R 33 which is composed of a rotation cos χ cos ω T cos χ sin ω T sin χ R jj = sin ω T cos ω T 0 sin χ cos ω T sin χ sin ω T cos χ and a boost β = β sin Ω T cos η cos Ω T sin η cos Ω T + β L sin ω T cos ω T where Ω is the earth s annual angular frequency and ω is the angular frequency of the sidereal day. The time T = 0 is defined to be when the earth crosses the celestial equatorial plane in the the year 2000 and the time T = 0 can be taken at any point when the laboratory y axis is aligned with the SCCEF Y axis. The angle between 98

115 5.4. ODD-PARITY ASYMMETRIC RESONATOR SENSITIVITY TO THE SME the earths orbital and equatorial planes is η and the co-latitude of the laboratory is χ. The boost component of Lorentz Matrix Λ μ ν is caused by the orbit of the earth around the sun. The value for the orbital boost is β = v /c 10 4 where v is the orbital velocity of the earth and c is the speed of light. The boost provided by the rotation of the earth at the surface β L is less than and zero at the poles. The relation of the κ matrices in the SCCEF to the laboratory matrices is given by [Kostelecký 02]: (κ DE ) jk lab = T jkjk 0 (κ DE ) jk T jkjk 1 (κ DB ) JK 5.8a (κ HB ) jk lab = T jkjk 0 (κ HB ) jk T jkjk 1 (κ DB ) JK 5.8b (κ DB ) jk lab = T jkjk 0 (κ DB ) jk + T jkjk 1 (κ DE ) JK + T jkjk 1 (κ HB ) JK 5.8c where T jkjk 0 = R jj R kk and T 1 = R jp R kj ɛ KPQ β Q and ɛ is the standard antisymmetric tensor. In a laboratory resonator based test of Lorentz invariance we are sensitive to some electrodynamic observable O. A deviation from the the conventional value of O caused by violations of Lorentz invariance is denoted δo and taken to be linear in the κ matrices. In the laboratory frame it can be written by δo =(M DE ) jk lab (κ DE) jk lab +(M HB) jk lab (κ HB) jk lab +(M DB) jk lab (κ DB) jk lab 5.9 where M DE,M HB and M DB are specific to the electrodynamics of the experiment. Rotation of an experiment in the laboratory provides a further modulation of the κ parameters relative to the SCCEF and the this is accounted by time varying M matrices related to the rotation. The experiments reported here are all nominally stationary in the laboratory reference frame so such derivations are unnecessary. 5.4 Odd-Parity Asymmetric Resonator Sensitivity to the SME The original Michelson-Morely test was an even-parity experiment and most modern tests of LI have retained this symmetry giving leading order sensitivity to the even- 99

116 CHAPTER 5. TESTS OF LORENTZ INVARIANCE parity SME parameters. Sensitivity to odd-parity and scalar violations of Lorentz invariance are suppressed in even-parity experiments. The first proposal to use an odd-parity experiment to test LI was presented in [Tobar 05], illustrating that an asymmetry in the experiment provided by the presence of a magnetic permeable material (in the case of an interferometer) or a dielectric (in the case of a ring resonator) in one arm the apparatus gives enhanced sensitivity to the odd-parity κ jk o+ parameter and the isotropic parameter κ tr. Based on [Tobar 05], and we will elucidate the advantages of odd-parity resonator. In any resonator experiment the observable is the fractional frequency shift, δν/ν 0 of the cavity resonance. To calculate the shift in resonant frequency we use equation 5.10, as derived in [Kostelecký 02]. δν = 1 ( ɛ 0 E ν 0 4 U 0c.κ DE.E 0c 1 [ ]) B ɛ0 V μ 0c.κ HB.B 0c + 2Re E 0 μ 0c.κ DB.B 0c dv where U = V (E 0c.D 0c + B 0c.H 0c)dV 5.11 U is the energy filling factor and serves as a normalization factor in equation From equation 5.9 the M matrices become: (M DE ) jk lab = ɛ 0 (E j 4 U 0cE k 0c)dV V (M HB ) jk lab = (M DB ) jk lab = Re 1 4μ 0 U [ 1 2 U (B j V ɛ0 0cB k 0c)dV μ 0 V (E j 0cB k 0c)dV ] 5.12a 5.12b 5.12c To calculate the sensitivity to the SME parameters it is only necessary to consider the (M DB ) jk lab coefficient of (κ DB) jk lab because this term gives rise to non zero and time varying κ o+ and κ tr in an asymmetric resonator. A generic asymmetric traveling wave ring resonator is shown in figure 5.3 with different permeability and permittivity and the two arms. For an asymmetric resonator the calculation of the sensitivity to the SME parameters involves the evaluation of integral 5.12c and the details depend on the fields in a specific experiment. The calculation for the asymmetric ring resonator used in 100

117 5.4. ODD-PARITY ASYMMETRIC RESONATOR SENSITIVITY TO THE SME Figure 5.3: A generic asymmetric ring resonator with relative permittivity ɛ ra and permeability ɛ ra in one arm and ɛ rb, μ ra in the other arm. this experiment will be presented in later chapters, but from the analysis the generic asymmetric resonator of figure 5.3 we obtain the following result [Tobar 05] 1 (M) xy lab (M)yx lab = ɛra μ ra ɛ rb μ rb 2+ɛ ra + ɛ rb If one arm of the resonator is vacuum and the other arm is a dielectric material with permeability 1, the index of refraction is n = ɛ ra and the above equation becomes: (M) xy lab (M)yx lab = ɛra 1 ɛ ra +3 = n 1 n and such a resonator is sensitive to the κ tr parameters and various κ o+ parameters, depending on the orientation of the experiment, and the result is independent of the polarization of the light. The important result of [Tobar 05] is the fact that an odd-parity asymmetric resonator is first order sensitive to the odd-parity κ o+ and only suffers a single factor of β ( 10 4 ) suppression of κ tr. This is in contrast to even-parity Michelson-Morely experiments that are first order sensitive to the even-parity κ e parameters, have β suppression of the κ o+ terms and suffer β 2 suppression of the isotropic parameter κ tr [Hohensee 10]. Therefore an odd-parity resonator will be 10 4 times more sensitive to odd-parity and isotropic violations of Lorentz invariance than an even-parity resonator with equivalent fractional frequency uncertainty, see table 5.2. It should be noted that there are other proposals for odd-parity tests of LI. 1 Note equations (34) and (36) in [Tobar 05] contain a minor typographical error. Source: M. Tobar, private communication 101

118 CHAPTER 5. TESTS OF LORENTZ INVARIANCE Table 5.2: resonators Approximate sensitivity to the SME Parameters for different parity Experiment Parameter Sensitivity (β 10 4 ) Even Parity Odd Parity κ jk e β κ jk o+ β 2 κ tr κ jk o+ β κ tr β κ jk e An approach in [Mewes 07] uses cavity geometry to break the symmetry and enhance the sensitivity. Electro-static and magneto-static experiments can also have enhanced sensitivity to the odd-parity parameters of the SME [Bailey 04]. The relative simplicity of an asymmetric ring resonator with a dielectric filled arm suggest that such a resonator has the greatest promise of improving the constraints on the κ o+ and κ tr parameters. 5.5 Experimental Constraints on the SME parameters All tests of Lorentz invariance are created as null experiments. The possibility of a violation of LI is assumed and the effect on the experiment is quantified: thus if the result is consistent with the noise floor of the experiment no effect is measured then no violations of LI is measured. All tests of LI must quote the uncertainty of the measurement, real experiments contain noise which limits the precision of the experiment and the constraints on the parameters. In the framework of the SME non-zero parameters indicate a violation of LI and the value of the parameter indicates the degree to which LI is violated. So far no violation of Lorentz invariance has been reliably detected Even-Parity Experiments Modern even-parity resonator tests of LI have rapidly improved the constraints on the κ e parameters by 4 orders of magnitude in less than a decade, see figure 5.4. An experiment stationary in the laboratory comparing a vertical microwave cavity with an East-West orientated cavity produced limits at the level [Lipa 03]. 102

119 5.5. EXPERIMENTAL CONSTRAINTS ON THE SME PARAMETERS Figure 5.4: Improvement of the constraints on the even parity coefficient ( κ e ) XY. The constraints on the odd-parity SME coefficients are 10 4 larger than the even parity coefficients. (a) [Lipa 03], (b) [Müller 03], (c) [Wolf 04], (d) [Müller 05], (e) [Stanwix 05], (f) [Müller 07], (g) [Herrmann 05], (h) [Stanwix 06], (i) [Eisele 09], (j) [Herrmann 09], (k) [Hohensee 10]. Results at the level were reported by an experiment using 90 orientation modulation of two orthogonal optical resonators [Antonini 05]. The comparison of a sapphire resonator to an H-maser [Wolf 04] and a measurement of the difference frequency from two orthogonal optical resonators [Müller 03] reduced the constraints to the level and considerations of light propagation in matter filled resonator experiments constrains some of the even coefficients at a similar level [Müller 05]. Experiments that rotate in the laboratory have multiple advantages - sensitivity to more SME coefficients and, the higher frequency of expected LI signals occur in regions with less noise. Using a rotating optical resonator referenced to a stationary optical resonator, sensitivity at the level was achieved [Herrmann 05] and a fully rotating microwave resonant cavity experiment had similar sensitivity after a year of data acquisition [Stanwix 06, Stanwix 05]. Comparisons between the rotating optical and microwave experiments allows for limits to be placed on more SME coefficeints than a single experiment [Müller 07]. The best constraints are provided by rotating orthogonal optical resonant cavities in monolithic fused sil- 103

120 CHAPTER 5. TESTS OF LORENTZ INVARIANCE ica [Herrmann 09] or ultra low expansion blocks [Eisele 09]. These experiments have constrained the even-parity κ e coefficients to the level. Due to the boost suppression of odd-parity SME parameters in even-parity experiments the constraints for the κ o+ coefficients from the previous experiments are 10 4 larger than the even parity constraints and are currently at the level Constraints on κ tr As an isotropic shift in the speed of light there are many experiments sensitive to the κ tr parameter ranging from resonator experiments to particle physics. Non-zero κ tr modifies the decay rates for two processes: the production of vacuum Cherenkov radiation and photon decay, based on assumptions of high energy dynamics. The absence of either process in quality high energy cosmic ray measurements give stringent bounds at the level [Klinkhamer 08]. In particle accelerators the modification of the limiting of velocity of photons and electrons from non-zero κ tr cause the generation of vacuum Cherenkov radiation and photon decay. Again the lack of either process gives constraints at the level [Altschul 09, Hohensee 09a, Hohensee 09b] and the characterisation of synchrotron losses at the level [Altschul 09]. The contribution of κ tr to the anomalous magnetic moment of the electron can be calculated and compared to the standard value, giving a giving a constraint at the 10 8 level [Carone 06]. However, the previously mentioned constraints on κ tr are deduced on theoretical grounds and contain model dependent assumptions or are indirect measurements of κ tr [Kostelecký 11]. Direct measurements of κ tr can be derived from the results of classical Ives- Stilwell type experiments [Ives 38] measuring time dilation factors. Spectroscopy on fast moving ions have set limits on the RMS parameter α, which represents a deviation from time dilation as specified by special relativity. The SME parameter κ tr can be obtained from these experiments by consideration of the phase velocity of signals traveling in opposite directions in the laboratory frame with respect to the SCCEF [Tobar 05] and the sensitivity of these tests to κ tr is at the 10 7 level for an analysis of [Saathoff 03] and reported at the 10 8 level in [Reinhardt 07]. 104

121 5.6. PROPOSED EXPERIMENT The current best direct constraint on κ tr is provided by an improved analysis of the data collected by [Stanwix 06], an even-parity rotating microwave experiment. Due to the even-parity of the experiment the sensitivity to κ tr is suppressed the boost β compared to the κ e parameter. The experimental determination of κ tr is 15 ± [Hohensee 10]. The only odd-parity test of LI reported is a rotating asymmetric microwave interferometer which was limited to 10 7 sensitivity by vibrational noise [Tobar 09]. Figure 5.5: Direct Measurements of the isotropic κ tr from Ives-Stilwell type experiments [Reinhardt 07] and even-parity microwave resonators [Hohensee 10]. 5.6 Proposed Experiment From the previous section it is clear that the odd parity κ o+, and especially the scalar κ tr are severely under-constrained compared to the even parity κ e. This is due to the wide use of even-parity Michelson-Morley type experiments in searches for violations of Lorentz invariance. Since all parameters of the SME are independent, the tight constraints placed on certain parameters have no implications for the size of the other parameters. Even-parity experiments have first order suppression of sensitivity to the boost dependent parameter κ o+ and second order suppression of sensitivity to κ tr. An oddparity resonator is leading order sensitive to κ o+ and has only first order suppression of κ tr. The proposed experiment is to perform the first odd-parity optical resonator 105

122 CHAPTER 5. TESTS OF LORENTZ INVARIANCE experiment and improve the constraints on isotropic shifts of the speed of light, represented in the SME by κ tr. 106

123 The near stillness recalls what is forgotten, extinct angles Georg Trakl ( ) 6 Stationary Odd-Parity Test of Lorentz Invariance Chapter Overview This chapter presents the results of the first odd-parity optical resonator test of Lorentz invariance. The use of counter-propagating modes gives the experiment high levels of immunity to environmental fluctuations and the experiment was able to improve the constraint of the isotropic standard model extension parameter κ tr. 6.1 Asymmetric Optical Resonators This experiment, the first odd-parity resonator test of Lorentz invariance, is based on an asymmetric optical ring resonator and in this section we will calculate the sensitivity of the experiment to the SME parameters. The asymmetry required by the odd-parity nature of the experiment is provided by the presence of a dielectric in one arm of the ring resonator. We use a Brewster angled UV fused silica (n=1.44 [Nikogosian 97]) prism as the dielectric medium and the resonator has been designed so that the light is incident at the Brewster angle, θ B = For the correct polarization of light (p-polarised) losses at the interface of the prism are minimized while for the other polarization (s-polarized) the losses are very large, see figure 6.1 for a diagram of the asymmetric cavity. The fractional change in resonant frequency due Lorentz violating SME parameters is given by equation 5.10 and from [Tobar 05] 107

124 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Figure 6.1: The asymmetric optical ring resonator. The dielectric is a Brewster angled UV fused silica prism with the resonator deigned so the the light is incident at Brewster s angle θ B we are only required to evaluate equation 5.12c, repeated below: [ (M DB ) jk lab =Re 1 ] ɛ0 (E j 2 U 0cB k 0c)dV and the electromagnetic filling factor U of equation μ 0 V 6.1 In all experiments the ring resonator was orientated with propagation in the dielectric in the East-West direction. resonator is a traveling electromagnetic wave given by: The resonant mode of an asymmetric ring E = E 0 e i(k.r ωt ) B = ɛ r ɛ 0 μ r μ 0 (k E) 6.2 where k is the direction of propagation, ω is the angular frequency of the electromagnetic wave. In the two arms of the asymmetric ring resonator there are materials with different relative permittivity ɛ r and relative permeability μ r. The permittivity and permeability of free space are ɛ 0 and μ 0 respectively. The orientation of the fields for vertical ring resonator is shown in figure 6.2. In the experiment the fused silica prism has relative permittivity ɛ r(fs) =2.07 and the vacant arm is filled with air ɛ r(air) 1, the relative permeability of each material μ r 1. Application of equation 6.1 to the configuration in figure 6.2 yields an (M DB ) jk lab matrix with only one non-zero entry: (M DB ) zx lab = L c ɛr +2L A cos φ m L v L c ɛ r + L v +2L 6.3 A which reduces to equation 5.14 in the case of an orthogonal resonator with equal arms lengths - the generic resonator shown in figure 5.3. As expected varying the 108

125 6.2. COUNTER-PROPAGATING MODES Figure 6.2: The orientation of the fields for vertical ring resonator with direction of light propagation in the dielectric in the East-West direction, using the standard co-ordinates of the SME. L c is the optical path length in the dielectric, L v is the parallel path in the vacant arm, L A is the angled path lengths and φ m is as shown. orientation of the resonator changes the non-zero entry in (M DB ) jk lab. For example a North-South light propagation in the dielectric changes the non-zero component to (M DB ) zy lab. There is a loss of sensitivity from the geometry of the cavity imposed by the Brewster s angle, maximum sensitivity is found when the optical path length L v in the vacant arm is equal to the dielectric path length L c and the sensitivity scales with the ratio of the two lengths. 6.2 Counter-Propagating Modes Repetition of the above analysis for an electromagnetic wave traveling in the opposite direction in the asymmetric ring resonator gives the result that the sign of the non zero (M) jk lab entry is reversed, and a violation of Lorentz invariance has the opposite effect on the resonant frequency of the counter-propagating mode in the asymmetric ring resonator. This indicates that by measuring the frequency difference between the resonance frequencies of two counter-propagating modes in an asymmetric ring resonator the experiment is sensitive to violations of Lorentz invariance. There are two advantages to such an approach. By taking the difference between two counter-propagating modes there is twice the sensitivity compared to an experiment with the resonant frequency of a single direction of propagation referenced to a linear cavity. 109

126 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Figure 6.3: Counter-propagating modes in an asymmetric ring resonator. A measurement of the difference in the resonance frequency (ν 2 - ν 1 ) is sensitive to Lorentz violating SME parameters The more important advantage from the use of counter-propagating modes is from an experimental standpoint. In resonator tests of Lorentz invariance the observable is the fractional change in the resonance frequency and this is indistinguishable from length fluctuations of the resonator. There are numerous causes of environmentally driven length fluctuations such as thermal expansion, variations in mechanical stress, vibration, material creep and changes in the refractive index of the prism. This can limit the sensitivity of the experiment to violations of LI and great lengths are taken to isolate the cavity from the environment and provide dimensionally stable resonators. Counter propagating waves with the same spatial mode, in the same resonant cavity, must necessarily have complete spatial overlap of the beams within the resonator and virtually all length changes in the resonator will be common to both directions. There is some miniscule, but non-zero sensitivity to optical path length fluctuations arising form perturbations affecting the counterpropagating modes differently as they transit around the cavity. Such extremely fast perturbations would be very quickly averaged away and are not considered a significant source of noise. The use of counter-propagating modes gives massive commonmode rejection of length fluctuations in the resonator when the difference frequency ν 2 - ν 1 is the observable. The experiment is still sensitive to the SME parameters, as the change in resonant frequency is opposite for the counter-propagating modes and will not be common mode rejected. The use of counter propagating modes in asymmetric resonator gives sensitivity to odd-parity and scalar violations of Lorentz invariance and extensive suppression of systematic effects. This allows a relatively 110

127 6.3. MODULATION FREQUENCIES AND THE SHORT DATA SET APPROXIMATION simple experiment to be highly sensitive to violations of Lorentz invariance. 6.3 Modulation Frequencies and the Short Data Set Approximation In the SME the Lorentz transformation from the SCCEF to the laboratory reference frame causes a modulation of the κ parameters that can be detected by experiment. For an experiment stationary in the laboratory the annual and sidereal rotations of the earth cause a modulation of the boost β from the orbital velocity of the earth β. In applying the transformation from the SCCEF to (κ DB ) jk lab (equation 5.8c) we ignore the rotational velocity of the earth because it is two oders of magnitude less than the orbital velocity. The other SME parameters besides κ o+ and κ tr are set to zero in the calculation as they constrained to many orders of magnitude smaller then the odd-parity and scalar parameters [Tobar 05]. The following tables are the modulation frequencies for an odd-parity, stationary experiment measuring the difference frequency of counter-propagating modes. The notation used to represent the M matrices is (M DB ) JK lab =(M DB) JK lab (M DB) KJ lab and the numerical character denotes a direction of the counter propagating modes, 1 is anti-clockwise propagation and 2 is clockwise propagation in figure 6.2 and we are measuring (ν 2 - ν 1 ). Sensitivity coefficients for an experiment rotating in the laboratory can be found in [Tobar 05]. From tables 6.1 and 6.2 the signal for violations of LI occurs at the rotation frequency for an odd-parity resonator test of LI, which is the sidereal frequency (plus annual modulation) for an experiment stationary in the laboratory. This is in contrast to an even parity experiment where the signal for violations if LI occurs at twice the rotation frequency. The different symmetries of odd and even-parity experiments experiments are the cause of the difference, even parity experiments are equivalent after rotating 180, odd-parity experiments require a full rotation. The disadvantage of odd-parity experiments is the equivalence of rotation frequency and LI effect, in even parity 111

128 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Table 6.1: Modulation frequencies for κ tr for a stationary experiment using counter-propagating modes. Modulation Coefficient cos(ω T +Ω T ) κ tr β (cos η 1) [(M2 DB ) XZ lab M1 DB) XZ lab ] sin(ω T +Ω T ) κ tr β (cos η 1)([(M2 DB ) YZ lab M1 DB) YZ lab ]cosχ +[(M2 DB ) XY lab M1 DB) XY lab ]sinχ) cos(ω T Ω T ) κ tr β (cos η +1) [(M2 DB ) XZ lab M1 DB) XZ lab ] sin(ω T Ω T ) κ tr β (cos η + 1)([(M2 DB ) YZ lab M1 DB) YZ lab ]cosχ +[(M2 DB ) XY lab M1 DB) XY lab ]sinχ) resonators systematic effects at the rotation frequency can be can be disregarded as the LI effect appears at twice the frequency The Short Data Set Approximation The short data set approximation is used in the case of data sets spanning only a small portion of the year [Stanwix 05]. Over the course of a short data set the angle of orbit Φ = Ω T in the SCCEF does not vary significantly. The change in the angle of orbit δφ Φ Φ 0 is small compared with respect to a full rotation and the double angle formula gives: cos(ω T ± (δφ+φ 0 )) = cos(ω T ± δφ) cos(φ 0 ) sin(ω T ± δφ) sin(φ 0 ) cos(ω T )cos(φ 0 ) sin(ω T ) sin(φ 0 ) sin(ω T ± (δφ+φ 0 )) = sin(ω T ± δφ) cos(φ 0 ) ± cos(ω T ± δφ) sin(φ 0 ) sin(ω T )cos(φ 0 ) ± sin(ω T ) sin(φ 0 )

129 Table 6.2: 6.3. MODULATION FREQUENCIES AND THE SHORT DATA SET APPROXIMATION counter-propagating modes. Modulation frequencies for κ XZ o+ for a stationary experiment using Modulation cos(ω T ) Coefficient κ XZ o+ [(M2 DB ) XZ lab M1 DB) XZ lab ] sin(ω T ) - κ XZ o+ ([(M2 DB ) YZ lab M1 DB) YZ lab ]cosχ +[(M2 DB ) XY lab M1 DB) XY lab ]sinχ) where Φ 0 is the average orbital angle when the short data set is taken. The short data set linearizes the orbital rotation and the ω T ± Ω modulation becomes ω T modulation with a constant orbital angle Φ 0, as shown in table 6.3. Table 6.3: Modulation frequencies for κ tr for a stationary experiment using counter-propagating modes and using the short data set approximation Modulation sin(ω T ) 2 κ tr β [(M2 DB ) XZ lab Coefficient M1 DB) XZ lab ] sin(φ 0) [1 κ tr β + κ tr β cos(η) + cos(η)] ([(M2 DB ) YZ lab M1 DB) YZ lab ]cosχ +[(M2 DB) XY lab M1 DB) XY lab ]sinχ)cos(φ 0) cos(ω T ) 2 κ tr β [(M2 DB ) XZ lab M1 DB) XZ lab ]cos(η)cos(φ 0) +[1 + κ tr β ( κ tr β 1) cos(η)] ([(M2 DB ) YZ lab M1 DB) YZ lab ]cosχ +[(M2 DB) XY lab M1 DB) XY lab ]sinχ) sin(φ 0) By searching for modulations of the form: ν 2 ν1 ν 0 = A s sin(ω T )+A c sin(ω T ) 6.5 in the resonance frequency difference between counter propagating modes an experimental determination of κ tr and κ YZ o+ can be made. 113

130 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ 6.4 Experimental Setup - Ring Resonator The physical realization of the asymmetric ring resonator analyzed in the previous sections is shown in figure 6.4(a) and the dimensions of the resonator are shown in figure 6.4(b). The cavity housing is precision machined from a single block of aluminium using CNC electric discharge machining and the optical path is drilled out of the block. The dielectric is an Optarius UV fused silica Brewster prism horizontally fastened to the aluminium block with a teflon clamp. The mirrors are housed in aluminium rings and there are two ring piezo-electrical actuators on one of the mirrors to allow cavity length adjustment. (a) Photograph (b) Dimensions Figure 6.4: The asymmetric optical ring resonator The mirrors are two identical low loss dielectric mirrors with a 100mm radius of curvature and 99.8% reflectivity. A linear Fabry-Perot cavity was constructed using the mirrors and the finesse was 1200 with a contrast of 0.3. The poor contrast of the linear cavity was attributed to losses in the mirrors, with the measured transmission 4 times smaller than expected from the reflectivity. The asymmetric ring resonator has a finesse of 860 and and this indicates the prism adds an extra to the round trip loss for the light. The losses caused by the presence of the Brewster angled prism are absorption, surface roughness and losses from a deviation from the optimal Brewster angle. The last mechanism is quite forgiving, a 1 misalignment only causes less than loss and the machining specifications are certainly better 114

131 6.5. EXPERIMENTAL SETUP - OPTICS AND FREQUENCY LOCKS than that. The additional losses are thought to arise form the surface roughness, the absorption of UV fused silica is extremely low [Nikogosian 97] and we find that losses are consistent with the 10/5 scratch/dig specification of the prism [Ridgway 98]. The contrast of the asymmetric ring resonator is 0.1 and the resonator is under-coupled, the transmission of the input mirror is not large enough to balance the losses in the resonator. The relatively high losses in the asymmetric ring resonator give wide resonances, the cavity bandwidth is ν cav = 4.2 MHz, and the free spectral range is set by the optical path length at ν FSR =3.6GHz. Due the presence of the Brewster angled prism the fundamental mode of the asymmetric ring resonator will be slightly astigmatic, with different waist sizes in the sagittal and tangential planes [Freegarde 01]. The astigmatism is more pronounced at the waist in the prism with a 134 μm waist and a 112 μm waist in the tangential and sagittal planes respectively. The waist in the vacant arm has suffers only minor astigmatism with 114 μm and 119 μm waists, a difference of only 4%. The input beam is mode-matched and coupled into the vacant arm by a 300mm lens and such a small astigmatism not significant, considering the inaccuracies in the mode matching process. By observation through the transmission port of the resonator it was verified that we are exciting the fundamental mode in both directions. This ensures that there is a complete spatial overlap of the counter-propagating modes and a rejection of optical path length fluctuations. Higher order modes are not frequency degenerate with the fundamental mode and will not be excited while the laser is locked to the fundamental mode. 6.5 Experimental Setup - Optics and Frequency Locks The Light Wave Electronic, 1064 nm laser from the previous chapters is used for these experiments. To detect violations of Lorentz invariance two independent measures of the resonant frequencies of the counter-propagating modes of the asymmetric ring resonator are required. Having only one laser at our disposal it was necessary to develop a scheme that enabled independent frequency locks on the two counterpropagating modes and a method to record the frequency difference of the two. An 115

132 CHAPTER 6. STATIONARY ODD-PARITY TEST OF LORENTZ INVARIANCE optical setup using two AOMs was employed, similar to the the scheme used for the two linear Fabry-Perot cavities of the previous chapters but with some important differences. After preparatory optics to give a well collimated Gaussian beam (described in section 3.3) the laser is split into two paths. Path 1 is frequency shifted twice by a double-pass AOM driven by a constant 80 MHz (ν fixed ) signal from a signal generator locked to a 10 MHz reference from an H-maser. After passing through a Polarization Maintaing (PM) single mode optical fibre and more preparatory optics described below, the beam is coupled into the asymmetric ring resonator and the reflected beam is used to generate the standard PDH frequency error signal. The demodulation is the same as the Fabry-Perot experiment, except the modulation frequency has been increased to 987 khz. The error signal is fed back to the piezoelectric actuator on the laser crystal, so the laser is locked 160 MHz below the resonant frequency of the CCW propagating mode, ν 1. Figure 6.5: Experimental setup for stationary test of Lorentz invariance. The frequency difference between the counter-propagating modes is measured though the correction signal sent to AOM2, ν cor =(ν 2 ν 1 )+ν fixed The second path of the laser, equal in intensity to the first, is frequency shifted ν cor twice by double-pasing through an AOM driven by a signal generator centered 116

133 6.5. EXPERIMENTAL SETUP - OPTICS AND FREQUENCY LOCKS at 80 MHz (with DC frequency modulation). Again the laser passes through PM optical fibre and preparatory optics, and is incident on the asymmetric ring resonator on the other side, in the opposite direction, to path 1 - see figure 6.5. The reflected signal is used to generate the PDH error signal and the correction signal is sent to the signal generator driving the second AOM. Thus the frequency shift ν cor sent to the second AOM is half the frequency feedback required to remain resonant with the CW mode (plus an 80 MHz offset), given the laser is already locked to the CCW resonance. Therefore ν cor =(ν 2 ν 1 )+ν fixed, the difference between the resonant frequency of counter propagating modes and violations of Lorentz invariance will appear in this signal. The difference frequency is recorded by a frequency counter and the double pass setup of the AOMs mean the recorded frequency must be doubled to attain the correct signal. The AOM2 correction signal ν cor, in which we search for violations of LI, does contain ν fixed but a long term measurement of the signal generator output indicates mhz stability over a 24 hour period. As signals for LI violations occur in the difference between the frequency of counter-propagating modes the absolute frequency of the resonant modes are of no interest. This allows the cavity to be locked to the laser by the ring piezo-electric actuators on the cavity. As the experiment is immune to length fluctuations caused by temperature changes, air currents or vibration is was not necessary to place the resonator in a vacuum system or provide vibration isolation. The resonator was securely mounted on the optical table and the experiment was carried out in atmosphere. The laser is delivered to the asymmetric ring resonator via two polarization maintaining optical fibres. As only p-polarized light will resonant in the cavity a half-wave-plate is used to create the correct polarization out of the PM optical fibre. Variations in the polarization of the light reaching the resonator will effectively vary the contrast of the resonator so a polarizer is placed in the optical path of both arms, variations of polarization will be converted to intensity fluctuations and the PDH frequency stabilization technique is nominally insensitive to such fluctuations. During pre-liminary testing a setup similar to the dual Fabry-Perot experiments 117

134 CHAPTER 6. STATIONARY ODD-PARITY TEST OF LORENTZ INVARIANCE of the previous chapters was constructed with one of the Fabry-Perot cavities replaced by the asymmetric ring resonator operating with a single propagating mode. This allowed the noise floors of the ring cavity frequency lock to be measured, see figure 6.6, and showed that the white noise floor is consistent with the previously discussed limiting factor - laser intensity noise at the PDH modulation frequency. The lower finesse of the asymmetric ring resonator increases the frequency noise for a given laser intensity noise compared to the Fabry-Perot cavities. In stationary tests of LI the signal appears with sidereal period and systematics at long timescales will generally dominate such fast frequency fluctuations which are random and will be averaged down. Figure 6.6: PSD of fractional frequency fluctuations in a single propagating mode of the asymmetric ring resonator (red). The noise floor of the ring piezo-electric actuators (black) limits the frequency stability in a measurement of one direction of propagation. It is common to both counter-propagating modes and will be common mode rejected in a test of LI, so the frequency stability will be limited by the noise floor of the frequency lock (blue). The noise floor of the detection system (green) is also shown. 118

135 6.6. INHERENT SYSTEMATIC EFFECTS 6.6 Inherent Systematic Effects It is important to consider potential systematic effects in an asymmetric ring resonator. There are effects inherent to the design of the asymmetric ring resonator that may cause an unwanted shift in the frequency difference of the counter propagating modes. If the frequency shift was to vary then this would limit the sensitivity of the experiment or could create false signals that might mask a violation of Lorentz invariance. In this section we cover the possible mechanisms for inherent systematic effects, in a subsequent section we will address drifts from technical sources The Sagnac Effect Comparing the frequency difference of two counter-propagating modes of a ring resonator is the basis of ring laser gyroscopes, utilizing the Sagnac effect [Post 67]. A rotational velocity of the apparatus causes a difference in the round trip time for counter-propagating light waves, and in the case of a resonator causes a relative shift of the resonant frequency for counter-propagating modes. The magnitude of the effect in a passive resonator is given by [Meyer 83, Post 67] Δf Sag = 4 A. Ω r λp L 6.6 where A is the area vector enclosed by the ring resonator, Ω r is the angular velocity vector, λ is the wavelength of the light and P L is the optical path length around the perimeter of the ring resonator. In the case of a stationary experiment the angular velocity is provided by the rotation of the earth, Ω r = Rad/s and the angular rotation vector points in a North-South direction. Due to the Sagnac effect if the asymmetric ring resonator was oriented horizontally in Perth, Western Australia (31 57 S E) there would be a fractional frequency split between counter-propagating modes of , which is equivalent to a κ tr at the level. This is a constant offset and will only represent a systematic effect if it fluctuates with a period equal to the rotation of the experiment. Fluctuations of this nature are expected to be small as the angular velocity of the earth, the dimensions or tilt of the experiment must 119

136 CHAPTER 6. STATIONARY ODD-PARITY TEST OF LORENTZ INVARIANCE change. The angular velocity of the earth is stable to the 10 5 level and tilt effects will be negligible. A 1 C temperature change in the laboratory will induce a relative change in the frequency split from the Sagnac effect Δf Sag corresponding to a systematic effect at the level in κ tr. Even though the Sagnac effect is orders of magnitude below the expected sensitivity of the experiment, the asymmetric ring resonator was carefully orientated so as to be insensitive to the Sagnac effect. The cavity was oriented vertically with the area vector pointing East-West, orthogonal to the rotation vector of the Earth. From equation 6.6 this make the dot product A. Ω r zero and the cavity is nominally insensitive to frequency shifts caused by the Sagnac effect. Figure 6.7: Diagram of the orientation of the area vector of the asymmetric ring resonator A, in relation to the Earths angular velocity vector Ω. The Area vector is pointing in an East-West direction so it is orthogonal to the earth s angular velocity vector Non-Reciprocal Effects Non-reciprocal, directional dependent effects will result in the counter-propagating modes of the asymmetric ring resonator having a different frequency. The Faraday effect is a pure rotation [Guenther 90] of the polarization of light caused by a magnetic field parallel to the direction of propagation. Counter-propagating electromagnetic waves will be rotated in opposite directions and this breaks the reciprocity of the ring resonator and can lead to frequency shifts between the resonant modes. To investigate the consequences of Faraday rotation and birefringence on the asymmetric ring resonator we make use of Jones calculus to represent the two orthogonal 120

137 6.6. INHERENT SYSTEMATIC EFFECTS polarizations of the resonant modes [Chernen kii 72]. The Transmission of the Brewster angled prism is: B = p x 0 0 p y 6.7 where p x and p y are the transmission of linearly polarized light, for the Brewster angled prism p x 1andp y =0.93. A rotation of the polarization from the Faraday effect is: R(θ) = cos(θ) sin(θ) sin(θ) cos(θ) 6.8 and θ = ν B L where ν is the Verdet constant, B is strength of the magnetic field and L is the length of interaction in the Brewster angled prism. The angle θ is opposite for the counter-propagating modes. Birefringence in the prism is given by: ( Bi(φ) = eiδx cos 2 (φ)+e iδy sin 2 (φ) e iδx + e iδy) cos(φ) sin(φ) ( e iδx e iδy) cos(φ) sin(φ) e iδy cos 2 (φ)+e iδx sin (φ) where φ is the orientation of the fast axis with respect to the x axis and δx and δy is the phase shift. The mirrors in the cavity are represented by: M = The polarization eigen-modes of the asymmetric ring resonator are the eigen-vectors of the equation for one round trip of the cavity: M.B.Bi(φ)R(θ).R.B.M 6.11 which has two very slightly elliptically polarized solutions. One is in the y axis, this is the s-polarized light and as no such light is incident on the resonator and the losses for this polarization are very large, this solution is discarded. The other solution is the p-polarized light propagating in the cavity. In the case of only Faraday rotation the two counter-propagating waves are elliptically polarized in opposite directions due to the non-reciprocal Faraday effect, but there is no net phase shift between them. In the case of only birefringence there is a phase shift of the two orthogonal polarizations but it is equal for the two counter-propagating 121

138 CHAPTER 6. STATIONARY ODD-PARITY TEST OF LORENTZ INVARIANCE directions so there is no net effect. In the presence of both Faraday rotation and birefringence there is an effect. The polarizations of the counter-propagating modes are oppositely rotated from the Faraday effect to give different elliptical resonant modes for the two directions of propagation. Each direction of propagation then acquires a different phase shift from the birefringence and a frequency shift between the counter-propagating modes can occur. The magnitude of the phase shift is dependent on the strength of the magnetic field parallel to the direction of propagation and the amount and direction of birefringence of the Brewster angle prism. The magnetic field strength next to the asymmetric ring resonator was recorded in the appropriate direction as 15 mt with less than 10 3 variability over a day. The stress birefringence is harder to measure (there is applied stress from the clamping of the prism) and we note that there is a dependence on the direction of the fast axis with respect to the p-polarised light. It was found theoretically that the non-reciprocal effect was maximized when the fast axis of the birefringence was at angle of π/8 Rad to the p-polarized light and the frequency shift is zero for the fast axis along the x and y axis because the two elliptic polarizations are symmetric about these axes. To measure the susceptibility of the experiment of non-reciprocal frequency shifts a magnet was placed with its field parallel to the direction of propagation in the ring resonator and orientation of the field was flipped multiple times. The strength of the magnetic field at an equal distance from the resonator was measured as 100 times stronger than the earth s magnetic field and no effect was observed with 1 khz peak to peak of random noise, which sets an upper limit on the stress birefringence in the crystal. The strength of the magnetic field is expected to vary more than temperature or mechanical driven changes in the stress birefringence. Using the measured strength of the magnetic field the polarization eigen-modes of the resonator are elliptical at the 10 6 level and a maximum fractional frequency split of was calculated. Variations in the magnetic field represent a systematic effect in a measurement of κ tr at the level, two orders of magnitude below the sensitivity of the experiment. 122

139 6.6. INHERENT SYSTEMATIC EFFECTS Mode Coupling The two counter propagating modes in the asymmetric ring resonator have nominally the same frequency and it is possible for each counter-propagating mode to couple to the other through scattering: this effect is readily observed in mirco-resonators [Kippenberg 02]. Micro-resonators exhibit strong mode coupling because of their small size (10s of μm), very high Q (> 10 9 ), multitude of nearly degenerate spatial modes and a geometry that couples much of the scattered light back into resonant modes. The asymmetric ring resonator has none of these properties, so the coupling between counter-propagating modes is expected to be small by comparison. In the case of the asymmetric ring resonator back-scatter from surface inhomogeneity and density fluctuations in the prism transfers power from the initially excited mode into the counter-propagating mode [Gorodetsky 00]. The asymmetric ring resonator has been designed so that there are no surfaces in the resonator orthogonal to the direction of propagation to minimize the coupling between counter-propagating modes.. If the mode coupling is strong and the scattering rate approaches the lifetime of the resonator then one observes a doublet in the place of the original mode, see figure 6.8. The new eigen-modes of the resonator are the symmetric and anti-symmetric superposition of the counter-propagating modes and it is no longer possible to excite counter-propagating modes independently. However, if the coupling is weak and no doublet is observed the counter-propagating modes can be excited independently, though a small portion of the propagating mode will be scattered into the counter-propagating mode. This merely reduces the sensitivity of the asymmetric ring resonator to violations of LI by the proportion of light scattered into the counter-propagating mode. In solving the coupling equation for the case of scattering from one mode only into the counter propagating mode, the splitting of the resonance is symmetric, see figure 6.8. This assumes a reciprocal coupling of the light ie the scattering from one mode into the other is equal to the scattering back into the original mode. Symmetric splitting will be be common mode rejected in the difference between the frequency of counter-propagating modes, it will only slightly broaden the resonances 123

140 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Figure 6.8: Splitting of a resonance into a doublet, in the case of strong coupling. of the two directions of propagation. Measurements of the asymmetric ring resonator show no evidence of a doublet structure. To search for evidence of scattering into the counter-propagating mode a single direction of propagation was excited and no counter-propagating mode was observed above the noise at the 0.5 % level, relative to the intensity of the excited mode. The consequence of scattering at this level would reduce the ability of the experiment to constrain κ tr by 1%, and the level of scattering is expected to be much lower. 6.7 Data Analysis and Results There is a relationship between the Lorentz violating parameters of the SME and the observable of the experiment, and perturbations of the observable are expected to occur at definite frequencies related to the motion of the experiment relative to the SCCEF. To determine these Lorentz violating parameters, or to place an upper constraint on their possible magnitude, we search for modulations in the observable at the appropriate frequencies. In this experiment the observable is the difference frequency between counterpropagating modes in an asymmetric ring resonator, and from tables 6.2 and 6.3 the modulations are synchronous with earth s sidereal phase. The difference in resonant frequency is recorded over many days and ordinary least squares regression is used 124

141 6.7. DATA ANALYSIS AND RESULTS to determine the amplitude and error of the SME parameters The East-West orientation of the asymmetric ring resonator gives only non-zero (M DB ) xz lab and the numerical calculations of the sensitivity coefficients (from section 6.1) are shown in table 6.4 for an experiment located in Perth, Western Australia (31 57 S E) and we use the short data set approximation. Modulation sin(ω T ) (a) Coefficient sin(φ 0 ) κ tr (b) Modulation cos(ω T ) Coefficient 0.14 κ XZ o+ cos(ω T ) cos(φ 0 ) κ tr Table 6.4: Sensitivity coefficients for the SME parameters κ tr (a) and κ XZ o+ (b) The stationary experiment ran for 45 days from the 19 th of November 2010 with about 35 days of usable data. Continuous data acquisition time ranged from 4 days to over two weeks. To reduce dead time and fast frequency fluctuations the frequency counter had a gate time of 10 seconds. We are concerned with modulations with sidereal period (24 h 56 m 4.1 s) so the data was averaged into 20 minute blocks to speed up the data analysis process. The time tags of the data are converted to sidereal phase and the data was differentiated [Tobar 10], this removes offsets and linear drift while leaving the the sinusoidal varying signals expected from violations of LI. The coefficients of table 6.4 were adjusted to take into account the differentiation. Drift in the data from technical sources leads to a f 1 power spectral density and the least squares regression used in the analysis assumes a white, f 0,power spectral density - in other words a normal distribution of data. Differentiation gives the data a white power spectral density. The processed data is shown in figure 6.9 and the power spectral density is shown in The technique used to perform the ordinary least squares regression is based on the technique used in [Stanwix 05, Stanwix 06, Stanwix 07] to search for violations of Lorentz invariance using microwave resonators. A matrix formulation of least squares regression is used [Draper 98] where the parameters β of a set of indepen- 125

142 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Figure 6.9: Processed time series data of the fractional difference frequency of counter-propagating modes in the asymmetric ring resonator Figure 6.10: Power spectral density of the time series data. The signals for violations of LI are expected at 1 sidereal day. dent variables X are to be estimated from observations Y. The signals we expect from violations of Lorentz invariance are contained in the model matrix X. The relationship between the matrices is: Y = Xβ + ɛ 6.12 where ɛ is the error of the fit. Matrix manipulation gives b, the least squares estimate of β as: b =(X T X) 1 X T Y The variance-covariance matrix V(b) of the parameter estimates is given by: 6.13 V(b) =(X T X) 1 σ

143 6.7. DATA ANALYSIS AND RESULTS where the variance σ 2 is calculated from the the standard deviation of the residuals r from the fit to the data. The off-diagonal elements of the variance covariance matrix gives the correlations between the parameter estimates and for independent parameters the relative value of the covariance should be small. The residuals of the fit are: r = Y Xb 6.15 The standard error of the fit parameters is given by the square root of the corresponding diagonal elements of the variance-covaraince matrix V(b). In the data we fit to an harmonic series with an offset and drift of the form: A + BT + C ω i cos(ω i T )+S ω i sin(ω i T ) 6.16 i and the application of the least squares regression will yield estimates and standard errors for modulations with a sidereal period. S ω The data was split into 2 day sections and from the amplitude of of C ω coefficients a calculation of The SME parameters is made, see figure 6.11 for determinations of κ tr. and Figure 6.11: Values of κ tr determined from ordinary least squares regression. A weighted average of all the data sets give the final values for the SME parameters. The values obtained for κ tr is: κ tr =3.4 ± No violation of Lorentz invariance was detected, with the magnitude of the parameter within the error of the experiment. However the experiment has marginally 127

144 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ improved the constraint on the isotropic shift in the speed of light from the previous best direct measurement of 15 ± , from an even-parity microwave resonator [Hohensee 10], and has a significantly smaller parameter estimate. For the odd-parity SME parameter κ XZ o+ the value obtained is 0.7 ± which is an order of magnitude above the best current limits [Eisele 09, Herrmann 09]. The uncompetitive constraint on κ XZ o+ compared to κ tr is due to the constraints being derived from two different experiments with different sensitivity (optical resonators [Eisele 09] and microwave resonators [Hohensee 10] respectively). 6.8 Limitations of Sensitivity The use of counter-propagating modes renders the experiment insensitive to length fluctuations of the asymmetric ring resonator and the inherent systematics are well below the sensitivity achieved by the experiment. However the experiment has not reached the limit imposed by the white frequency noise floor of the PDH locking scheme. A possible limitation on the sensitivity is the presence of an offset in the error signal and variations in mode matching. The control electronics lock to the zero crossing so the combination of an offset in the error signal and changes in mode matching modify the height of the discriminator and cause the locking point to change. An offset in the error signal is caused by residual amplitude modulation of the laser from the PDH modulation de-modulated to DC and small electronic offsets present in the mixers. To maximize the common mode rejection of variations in the offset identical mixers are used for the demodulation and the two error signals of the counter-propagating modes have the same phase relative to the PDH modulation, which is common to both counter-propagating directions. From data recorded during the operation of the experiment, long term drift of 0.1 Hz/s in the difference frequency of the counter-propagating modes is correlated with the discrepancy in mode matching of the two directions, see figure The amount of discrepancy is inferred from measurements of the power of reflected light from the asymmetric ring resonator. The variability in the intensity of the laser is 10 times smaller than the observed drift in the reflected power so we attribute the 128

145 6.8. LIMITATIONS OF SENSITIVITY Figure 6.12: Exaggerated DC offset in a PDH error signal caused by residual amplitude modulation of the laser by the PDH frequency modulation changes in reflected power to unequal mode matching between the two directions. Random pointing fluctuations in each of the input beams will cause the alignment of the laser to vary and the mode matching to the fundamental mode will change. When the laser is locked to the fundamental mode changes in the mode matching will cause the amount of reflected light to vary as misalignment reduces the excitation of the fundamental mode. Light not coupled into the fundamental mode will be reflected and higher order modes will not be excited as they are not frequency degenerate with the fundamental mode. To model this effect a mode-matching parameter A m is added to the cavity equations from section 2.1 representing variations in the mode-matching [Dawkins 08]. The equation for the reflected signal becomes: F (ω) = r(a m A m r 2 + r 2 e ω ν FSR ) 1 r 2 e ω 6.18 ν FSR and we can repeated the analysis of section 3.1 to obtain the error signal, with an offset added in the error signal. From the model variations in the mode matching change the height of the discriminator shown in figure The amount of drift in the difference frequency is consistent with the drift calculated from model using the measured offset and mode-matching variability. The drift is not synchronous with the sidereal phase, it is essentially linear over several days and does not create a systematic effect in the test of Lorentz invariance. The long term drifts are converted to constant offsets by the differentiate of the data for analysis. Minor variations in the rate of the drift 129

146 CHAPTER 6. INVARIANCE STATIONARY ODD-PARITY TEST OF LORENTZ Figure 6.13: An example of long term frequency drift in the unprocessed fractional frequency difference between the two counter propagating modes (bottom), and the fractional variation in mode matching inferred from the reflected photodiodes (top). increase the standard deviation of the data and reduce the ability of the experiment to place constraints on the SME parameters. 6.9 Conclusions The use of counter-propagating modes in the odd-parity asymmetric ring resonator has exceptional suppression of environmental fluctuations, enabling a highly sensitive test of Lorentz invariance from a relatively simple experiment. The presence of the Brewster angled prism had negligible effect on the finesse of the resonator and has the advantage of minimizing retro-reflections. While there are some inherent systematic effects associated with the counter-propagating modes they are orders of magnitude below the sensitivity of experiment. After 45 days of data acquisition using the rotation of the earth for modulation the experiment was able to slightly improve the constraint on the isotropic SME parameter κ tr, with a significant reduction in the parameter estimate. The limitation of the sensitivity was found to be long-term minor mode-matching variations over the course of the experiment. 130

147 Marvelous Truth, confront us at every turn, in every guise Denise Levertov ( ) 7 Orientation Modulated Odd-Parity Test of Lorentz Invariance Chapter Overview To improve the performance of the experiment orientation modulation of the asymmetric ring resonator is employed. By periodic reversal of the orientation of the apparatus the signal for a violation of Lorentz invariance occurs on a much shorter timescale and the disadvantages of the previous experiment are overcome. This experiment was able to further improve the constraint on κ tr by a factor of six. 7.1 Orientation Modulation In general to increase the sensitivity of resonator tests of Lorentz invariance an experiment can be continuously rotated in the laboratory. Rotation increases the frequency of signals for violations of Lorentz invariance up to timescales where the resonators have higher stability. Rotation also gives sensitivity to more SME parameters than an experiment stationary in the laboratory. For the asymmetric ring resonator continuous rotation was evaluated as unfeasible due to the possibility of the Sagnac effect and the complications of a polarization maintaining rotating optical fiber coupler. However, orientation modulation in the lab can still be used to improve the sensitivity of the asymmetric ring resonator. The isotropic shift in the speed of light κ tr is modulated by the sidereal rotation of the earth with respect to the boost β caused by orbital motion. If the experiment 131

148 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE is rotated 180 on a timescale short compared to the rotation of the earth, then the orientation with respect to the boost β is reversed. The effect of non-zero Lorentz violating SME parameters on the observable of the experiment is also reversed, so the expected sidereal modulation of the SME parameter becomes flipped by the 180 rotation of the experiment, see figure 7.1. In the language of the SME, for the asymmetric ring resonator then the non-zero entry of (M DB ) jk lab becomes -(M DB) jk lab from a 180 rotation of the experiment, if the observable of the experiment remains (ν 2 ν 1 ). By periodical reversal of the orientation of the experiment, followed by stationary data acquisition time the signal for violations of Lorentz invariance is modulated at a higher frequency and there is no additional Sagnac effect. orientation modulation of the asymmetric ring resonator is analogous to the optical chopping technique, where a laser is intensity modulated to move the signal from DC to higher frequencies. The (a) (b) Figure 7.1: Orientation modulation of the experiment. A back-and-forth 180 rotation of the experiment (a) in the laboratory will cause the signal for violations of Lorentz invariance to have the form shown in (b) for ν 2 ν 1. To achieve an increase in the sensitivity of the experiment the modulation of the SME parameters should be at a period where the experiment has optimal stability. A SRAV of a section of data from the previous experiment is shown in figure 7.2, showing the better frequency stability of the resonator at short timescales. Alignment effects introduced by the rotation are eliminated by active alignment control of the mode matching to the asymmetric ring resonator. 132

149 7.2. ROTATION STAGE Figure 7.2: SRAV of a section of data from the stationary test of LI showing the better stability of the experiment at shorter timescales. By modulating the orientation of the experiment the signals for LI will occur at a higher frequency. 7.2 Rotation Stage To enable the rotation of the asymmetric ring resonator the cavity and reflection photodiodes are mounted on a 300mm 400mm optical breadboard attached to the rotation stage. The same frequency locking scheme from section 6.5 is used and as before, where the two counter-propagating modes are independently locked with two acousto-optic modulators. The laser is delivered to the breadboard through polarization maintaining optical fibres after the AOMs. The optical fibres and frequency control signals are positioned along the axis of rotation and are securely fastened to minimize flexure during the back-and-forth motion of the rotation stage. The rotation stage is a Newmark Systems RT-5-10 with a computer controller to drive the stepper motor. The rotation stage has an accuracy of 60 arc-seconds, a uni-directional repeatability of 5 arc-seconds and is easily capable of supporting and rotating the experimental setup. The use of counter-propagating modes in the asymmetric resonator renders the asymmetric ring resonator insensitive to unwanted effects caused by the vibration and tilt of the rotation stage during operation. These effects cause the dimensions of the resonator to vary but by taking the difference frequency of spatially degenerate counter-propagating modes there is common mode rejection of cavity length fluctu- 133

150 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE Figure 7.3: Photograph of the optical breadboard mounted on the rotation stage. ations. Variations of the tilt of the experiment may change the size of the Sagnac effect but the frequency split is an order of magnitude below the expected sensitivity of the experiment. The tilt of the optical table is 0.2 from vertical and the wobble of the rotation stage was below the noise of the tilt sensor at 0.04, so rotation of the experiment will create a negligible change in the Sagnac effect between the two orientations. Vibration of the experiment during the movement phase can interfere with the modulation of alignment for active control. To reduce vibration from the rotation stage reaching the optics a Sorbothane sheet was placed between the rotation stage and the optical breadboard and nylon screws were used to secure the breadboard. An accelerometer was placed on the edge of the optical breadboard and the vibration was measured during rotation, the results are shown in figure 7.4. The rotation stage exhibits large mechanical resonances at approximately 3 deg/s so the rotation speed was kept below this speed. The measurement of the vibration spectrum in figure 7.4 indicates the regions of lowest vibrational noise where the modulation of the alignment for active control would be best suited. 134

151 7.3. ALIGNMENT CONTROL Figure 7.4: The vibration of the rotation stage stationary (blue) and rotating (red), measured with an accelerometer. 7.3 Alignment Control The rotation of the experiment can cause variation in the alignment of the input beam to the asymmetric ring resonator. From section 6.8 it can be seen that misalignment will cause a shift of the frequency lock that can reduce the sensitivity of the experiment. To maintain optimal alignment of the input beams active alignment control was introduced [Dawkins 08]. The output couplers of the PM optical fibres delivering the laser to the rotation table were housed in 3-axis piezo-electrically adjustable optical mounts (Thorlabs KC1-PZ). The x and y tilt of the mounts are modulated with a 90 phase shift between the two axis by a Direct Digital Synthesis (DDS) signal generator (Novatech 409B). As the input beam is mis-aligned the amplitude of the modulation in the reflected signal becomes larger and the sign of the modulation gives the relative direction. The reflected photodiode signal is sent to a dual phase lock in amplifier (Femto LIA-MVD-200-H) with the reference signal from the DDS function generator and the demodulated in-phase and quadrature signals give the mis-alignment in the x and y directions. The error signal is integrated and summed onto the modulation frequency then amplified by the piezo driver. A schematic of the alignment control is shown in figure 7.5. To ensure optimal performance of the alignment control active power control was used to keep the power on 135

152 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE the reflected photodiodes constant by varying the RF power sent to the AOMs. Figure 7.5: Diagram of active alignment control. The DDS sends two modulation signals 90 out of phase to the x and y axes of the piezo mount and the lock in amplifier. Misalignment of the input beam is apparent in the reflected photodiode signal by demodulation from the lock-in at the modulation frequency. The error signal is integrated and summed to the modulation, completing the control loop. An identical system with a different modulation frequency is on the other input direction. To ensure stable operation the modulation frequencies for the two alignment controls were chosen away from the resonances of the piezo mount and the vibration peaks shown in figure 7.4. The modulation frequency chosen was 417 Hz and 479 Hz for the two counter propagating modes. The mechanical resonance of the pizeo mount is at 108 Hz and the bandwidth of the control loop was set below this to 10 Hz. The rotation of the experiment did not produce large alignment changes and the frequency control remained locked without the use of the active alignment control, see figure 7.6. When engaged the alignment control did reduce the long term drift of the experiment significantly and ensured systematic effects from the motion of the experiment did not contribute to the test of Lorentz invariance. 136

153 7.4. EXPERIMENTAL OPERATION Figure 7.6: Disengagement of one the alignment controls. The reversal of the experiment is evident on the alignment error signal (blue) when the control loop is disengaged, and the effect on the frequency difference of the two counter-propagating modes (red) is shown. 7.4 Experimental Operation At the beginning of the experiment the orientation of the apparatus is noted to ensure consistency during data analysis. During the stationary period data acquisition occurs for 617 seconds, a number chosen to avoid modulation synchronous with the sidereal day. To reverse the orientation of the experiment the rotation stage accelerates slowly at 0.2 deg/s 2 to a maximum angular velocity of 1.8 deg/s before decelerating at 0.2 deg/s 2 to an orientation rotated 180 from the previous one, and the 180 rotation takes 100 seconds. Another 617 seconds of stationary data is acquired and then the rotation stage reverses the rotation to return to the original orientation. The computer controller of the rotation stage outputs a high signal (+5 V) when the stage is moving and this signal is used in the data analysis to determine the changes in orientation of the experiment. An example of the experiment in operation is shown in figure 7.7. The flexing of the PM optical fibres from the rotation of experiment could vary the polarization of the light through the fibre, though ideally in PM fibres this is not the case. A polarizer in the optical path converts changes in polarization to power fluctuations that are removed by the power control system. 137

154 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE Figure 7.7: Recording the motion of the experiment. The move voltage (blue) is high when the rotation stage is in motion so successive lows represent a reversal of the orientation of the experiment. The fractional frequency difference of the two counter-propagating modes is also shown (red). The experiment is stationary for 617 seconds and the reversal of orientation takes 100 seconds. In the orientation modulated experiment the asymmetric ring resonator is oriented with the plane of propagating horizontal in the x-y plane, but the the propagation in the dielectric is still oriented in the East-West direction during the data acquisition phase. The change of ring resonator positioning was done to enable the optics to be mounted on low mounts to reduce any movement caused by rotation. The asymmetric ring resonator is now sensitive to the Sagnac effect from the rotation speed of the experiment in the laboratory, but the Sagnac effect only occurs during rotation and data is acquired while the experiment is stationary. The Sagnac effect during rotation is at the fractional frequency level and is below the short term noise of the experiment so no effect is observed. The reversal of the orientation of the asymmetric ring resonator reverses the non-reciprocal effects for Faraday rotation and birefringence. The direction of propagation of the two counter-propagating modes with respect to the magnetic field is reversed so the frequency splitting is modulated with the rotation of the experiment. The magnitude of the non-reicprocal splitting is but this is a constant offset between the two orientations. To affect the measurement of κ tr the magnetic field 138

155 7.5. DEMODULATION AND RESULTS strength must vary and measurements in the laboratory show the magnetic field is constant to 10 3 which corresponds to a measurement of κ tr at the level. 7.5 Demodulation and Results The data acquired during the orientation modulated test of Lorentz invariance requires demodulation before the ordinary least squares analysis. To demodulate the data the following procedure is carried out: Stationary periods of data acquisition were identified through the recorded move voltage of the rotation stage and data taken while the apparatus was motion were discarded. An average of the frequency difference for the counter-propagating modes was taken over the stationary period to create a single data point ν i. The difference of the average frequency from successive orientation reversals is obtained by (ν i ( 1) i ν i+1 )/2 from 1 to N, the number of reversals. This demodulates the data and returns the signal for violations of Lorentz invariance to a sidereal period. To verify the demodulation technique a data set with a simulated signal for a violation of Lorentz invariance was generated and the demodulation technique returned the correct amplitude and phase of the original signal. Over 66 days from the 19 th of May 2011 the experiment reversed orientation 6129 times during 50 days of data acquisition. The demodulated data from the experiment is shown in figure 7.8. The demodulation technique returns the signal for violations of Lorentz invariance to the sidereal period. To search for violations of Lorentz invariance the data analysis from section 6.7 is repeated with ordinary least squares regression used to determine the amplitude and error of signals for Lorentz violation. The values obtained for κ tr are shown in figure 7.9. The power spectral density of the residuals from the fits are shown in figure 7.10, and the power spectral density is sufficiently white around the sidereal modulation frequency to justify the use of ordinary least squares. The weighted average of the data sets gives a determination of κ tr as: κ tr =0.3 ± This constraint is 6 times better than the results of the previous stationary test 139

156 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE Figure 7.8: Demodulated time series data of the fractional frequency difference of counter-propagating modes in the asymmetric ring resonator from the orientation modulated experiment. Figure 7.9: Values of κ tr determined from ordinary least squares regression for the orientation modulated experiment. of Lorentz invariance and the best reported constraint on κ tr. The determination of the odd-parity parameter gives κ XZ o+ =1.6 ± , a factor of two from the current constraints. The uncertainty of the measurement is consistent with the frequency noise floor imposed by the PDH locking system (see figure 6.6) averaged down over the course of the measurement. This indicates that the experiment has reached the limit of sensitivity for a measurement of this duration. 140

157 7.6. FUTURE ODD-PARITY TESTS OF LORENTZ INVARIANCE Figure 7.10: Power spectral density of the residuals from the ordinary least squares. 7.6 Future Odd-Parity Tests of Lorentz Invariance The are many possibilities to improve the sensitivity of odd-parity tests of Lorentz invariance. For the experiments mentioned above the measurement was ultimately limited by the noise white floor of the PDH locking scheme at / Hz. The level of this noise is a consequence of the relatively low finesse and poor contrast of the asymmetric ring resonator. The mirror choice made in the construction of the asymmetric ring resonator has led to the mirrors providing a large proportion of losses in resonator. Mirrors with higher reflectivity could increase the finesse, but more importantly mirrors with less loss and higher transmission would give the resonator better contrast. This would significantly improve the frequency discriminator slope and reduce the frequency noise floor of the experiment. Another avenue for improvement is a higher PDH modulation frequency, which would increase the frequency discriminator slope (see figure 3.3) and reduce the frequency noise floor of the locks. This could be achieved with an electro-optic modulator for fast phase modulation. For new odd-parity tests of Lorentz invariance, the gain in sensitivity from rotation is feasible with sufficiently sophisticated experiments. A bow-tie asymmetric ring resonator (shown in figure 7.11) would be sensitive to the odd-parity and scalar SME parameters. With suitable geometry the Sagnac effect can be cancelled be- 141

158 CHAPTER 7. ORIENTATION MODULATED ODD-PARITY TEST OF LORENTZ INVARIANCE tween the two enclosed areas and the experiment can be rotated in the laboratory. It would require temperature control or the use of ULE to maintain the cancellation of the Sagnac effect from temperature fluctuations and the rotation speed would have to be accurately monitored. Magnetic shielding would need to be employed to reduce the influence of non-reciprocal effects. For a Brewster angled prism a rotating experiment would require a polarization maintaing rotating optical coupler but such devices exist [Kajioka 89]. Figure 7.11: Bow-tie resonator for a rotating test of Lorentz invariance. With a suitable geometry it should be possible to cancel the Sagnac effect for a rotating experiment. Of course any experiment requires detailed analysis before undertaking and a new odd-parity test of Lorentz invariance is no exception, other configurations of odd-parity resonators may exist that are insensitive to the the systematic effects in a Brewster angled asymetric ring resonator. With a high finesse, well coupled resonator and reduced systematics an odd-parity test of Lorentz invariance could approach the sensitivity of state of the art even-parity experiments [Eisele 09, Herrmann 09] at the level. This would create a 10 4 improvement of the constraints on the odd-parity and scalar SME parameters and further the search for violations of Lorentz invariance started over 120 years ago. 142

159 If I have seen further it is only by standing on the shoulders of giants Isaac Newton ( ) 8 Conclusion 8.1 Sapphire Fabry-Perot cavities We have successfully implemented an optical frequency reference for use in the laboratory, based on room temperature linear sapphire optical resonators. The experiment served as a test bed for the odd-parity test of Lorentz invariance, providing significant insight into precision frequency metrology and leading to the development of sophisticated frequency control and noise characterisation techniques. While the performance of the sapphire Fabry-Perot cavities was satisfactory for the optical frequency metrology measurements undertaken during this project, the drawbacks of the non-zero coefficient of thermal expansion and the relatively low finesses, combined with the recent advances in ultra-high finesse ULE cavities, do not favor the continuing future of room temperature sapphire cavities as an optical frequency reference Temperature Control The stringent requirements for temperature control of a Fabry-Perot cavity with a non-zero coefficient of thermal expansion have been met through the use of multiple stage AC temperature control and large heat shields. From measurements of the thermal characteristics of the experiment we have shown excellent thermal isolation of the Fabry-Perot cavities with external equilibrium times approaching two days. The measurements of thermal characteristics revealed the importance of internal heat flow in the temperature control of macroscopic room temperature objects. The temperature variation of the cavities is just 3 nk over a second, which corresponds 143

160 CHAPTER 8. CONCLUSION to fractional frequency stability of the Fabry-Perot cavities of However the long term stability of the temperature control system was insufficient for the original intended purpose, that is to provide a reference cavity for the asymmetric ring resonator. It was found that temperature control of macroscopic object at room temperature is fraught with difficulty due to thermal gradients and the slow dynamics of heat flow. Ultimately the lower than expected performance necessitated a new approach to the LI project and led to the major innovation of counter-propagating modes in the asymmetric resonator Frequency Control The Pound-Drever-Hall technique was used to lock a laser to the Fabry-Perot resonators. We used acousto-optic modulators to provide two independent frequency control systems from a single 1064 nm laser. The performance of the acousto-optic modulators for high bandwidth frequency control was rigorously tested and no additional noise above 1 Hz/ Hz was identified. The use of acousto-optic modulators removes the requirement for an optical beat note for frequency comparison and greatly simplifies the experiment. This results indicates the viability of the technique for other optical frequency metrology experiments Performance of the Frequency Reference A systematic evaluation of noise sources in the Fabry-Perot frequency reference led to progressive improvements of the experiment. The limits of control bandwidth were overcome with the use of acousto-optic modulators for supplementary fast frequency control. It was confirmed that vibration of the Fabry-Perot resonators limits the stability of the experiment at lower frequency. Active vibration control or improved cavity mounting can reduce the effect of vibration on the sapphire Fabry- Perot resonators. The presence of laser intensity noise at the PDH modulation frequency limited the fractional frequency stability to the level of / Hz at 1 khz. Attempts were made to reduce this noise through the use of acousto-optic modulators for fast frequency control utilizing a second order filter. The fundamental 144

161 8.2. ODD-PARITY TEST OF LORENTZ INVARIANCE noise of the Fabry-Perot cavities are within an order of magnitude of the frequency stability acheived, however technical noise sources, magnified by the relatively low finesse of the Fabry-Perot cavities, prevented the experiment from reaching the fundamental limit. The use of higher reflectivity mirrors to increase the finesse of the cavities would be an immediate, straight forward and easily implemented solution to overcome the limitations from the technical sources of noise and improve the short term frequency stability. The Fabry-Perot cavities were used as a frequency reference to measure the stability of an iodine filled hollow core photonic crystal fibre, and easily surpassed the frequency stability required for the comparison. 8.2 Odd-Parity Test of Lorentz Invariance We used an asymmetric ring resonator to perform the first odd-parity optical test of Lorentz invariance. Odd-parity experiments have enhanced sensitivity to odd-parity and isotropic violations of Lorentz invariance compared to previous even-parity experiments. The use of counter-propagating modes in the asymmetric ring resonator gives high immunity to environmental fluctuations and enabled highly sensitive experiments from relatively simple experiments. The major advances of an asymmetric ring resonator and counter-propagating modes presented here give odd-parity optical resonators the capacity to perform exceptionally sensitive experiments of tests of Lorentz invariance in the future. For example, odd-parity experiments with intrinsic frequency stability similar to that of current even-parity Michelson-Morley type experiments would deliver an improvement of four orders of magnitude in experimental sensitivity to odd-parity and isotropic violations of Lorentz invariance. Increasingly sensitive tests of Lorentz invariance have been performed since before the advent of special relativity and with the development of the asymmetric ring resonator the groundwork has been provided for this to continue. Such progress will ultimately heighten our understanding of the universe, either through increased confidence in current theories or the discovery and verification of new physics. 145

162 CHAPTER 8. CONCLUSION Asymmetric Ring Resonator An odd-parity test of Lorentz invariance requires some degree of asymmetry. The asymmetry in the optical ring resonator was provided by a Brewster angled UV fused silica prism in one arm of the resonator. The Brewster angle of the prism means there are no surfaces orthogonal to the optical path to retro-reflect light and diminish the sensitivity of the experiment. The Brewster angle also minimizes losses in the cavity for p-polarized light and the inclusion of the prism only created additional losses equal to the losses already present from the choice of mirrors Counter-Propagating Modes To detect violations of Lorentz invariance we measure the frequency difference in counter-propagating modes of the asymmetric ring resonator. This approach remains sensitive to violations of Lorentz invariance while giving complete commonmode rejection of optical path length fluctuations. Unwanted length fluctuations must be suppressed and usually the elimination of such length fluctuations requires extensive isolation of the resonator from the environment and increases the complexity of the experiment. Counter-propagating modes provide a highly sensitive experiment free from the complexity of other resonator tests of Lorentz invariance. The inherent systematic effects from the use of counter-propagating modes are found to be 3 orders of magnitude below the achieved sensitivity of the experiment. The limitations to the experiment were found to be generated by technical, not fundamental noise sources and could be further reduced. Thus the use of counterpropagating modes in the asymmetric ring resonator is presents a viable and highly advantageous technique in the quest for increasingly sensitive resonator based tests of Lorentz invariance Stationary Odd-Parity Test of Lorentz Invariance A measurement was performed over 45 days, recording the difference in frequency of counter-propagating modes of the asymmetric ring resonator. The experiment was 146

163 8.2. ODD-PARITY TEST OF LORENTZ INVARIANCE stationary in the laboratory and relied on the rotation of the earth to modulate the experiment: any violations of Lorentz invariance would appear synchronous with the sidereal motion of the experiment. No statistically significant violation of Lorentz invariance was detected and the results of the stationary experiment was able to marginally improve the experimental constraints on the Lorentz violating isotropic shift in the speed of light, κ tr, see figure 8.1. While only a small improvement on the previous constraint, the result showed the potential of the asymmetric ring resonator in a relatively simple experiment Orientationally Modulated Odd-Parity Test of Lorentz Invariance To overcome the limitations of the stationary experiment an orientationally modulated experiment was performed. Periodic reversal of the orientation of the asymmetric ring resonator effectively modulates the signal for Lorentz violation at much higher frequency than the rotation of the earth. This allowed the test of Lorentz invariance to utilize the better short term stability of the asymmetric ring resonator while avoiding the complications of rotation. Any possible systematic effects from the periodic reversal of the experiment were mitigated by the use of active alignment and optical power control. From over 6000 reversals of orientation over 66 days we are able to improve the constraint on the possible isotropic shift in the speed of light by a factor of 6, see figure 8.1. The improved constraint confirms that a fundamental theory of the universe, Lorentz invariance, represents the behavior of reality at a level never measured before. 147

164 CHAPTER 8. CONCLUSION Figure 8.1: Comparison of direct measurements of κ tr. The stationary odd-parity test of Lorentz invariance marginally improved the constraint on κ tr and has a significantly smaller parameter estimate. The orientation modulated experiment further improved the constraint on κ tr by a factor of six. 148

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179 A Appendix Testing Lorentz Invariance Using an Odd-Parity Asymmetric Optical Resonator 163

180 RAPID COMMUNICATIONS PHYSICAL REVIEW D 84, (R) (2011) Testing Lorentz invariance using an odd-parity asymmetric optical resonator Fred N. Baynes,* Andre N. Luiten, and Michael E. Tobar Frequency Standards and Metrology, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (Received 6 April 2011; published 6 October 2011) We present the first experimental test of Lorentz invariance using the frequency difference between counter-propagating modes in an asymmetric odd-parity optical resonator. This type of test is 10 4 more sensitive to odd-parity and isotropic (scalar) violations of Lorentz invariance than equivalent conventional even-parity experiments due to the asymmetry of the optical resonator. The disadvantages of odd-parity resonators have been negated by the use of counter-propagating modes, delivering a high level of immunity to environmental fluctuations. With a nonrotating experiment our result limits the isotropic Lorentz violating parameter ~ tr to 3:4 6:2 10 9, the best reported constraint from direct measurements. Using this technique the bounds on odd-parity and scalar violations of Lorentz invariance can be improved by many orders of magnitude. DOI: /PhysRevD PACS numbers: Cp, Ft, Da I. INTRODUCTION The assumption of Lorentz invariance (LI) is a vital foundational component of modern physics. This fundamental symmetry of space-time has been rigorously tested since it was first postulated over 100 years ago and all such experiments have so far verified the standard model of particle physics and general relativity to within their precision. Nonetheless, the emergence of quantum gravity and other unified theories [1 4], which hint at possible LI violations, continue to give new impetus to undertake ever more precise tests of LI. In order to compare the quality of various experimental tests of LI one can make use of the framework of the minimal Standard Model Extension (SME) by Kostelecky and coworkers [5], which parameterizes all possible LI violations by known fields. If an experiment generates a nonzero parameter in this framework then it indicates the degree to which LI is violated. Although the SME is a comprehensive theory with particle, gravity and photon sectors, the experiment reported here is focused on the photon sector in which there are 19 independent parameters of the SME. In this sector astrophysical observations have determined that the 10 parameters representing vacuum birefringence, (~ jk eþ and ~ jk o ) are below [6]. The remaining anisotropic parameters ~ jk e and ~ jk oþ, as well as the isotropic parameter ~ tr, have been constrained through laboratory tests using optical or microwave resonators. The current constraints on the anisotropic evenparity (~ jk e ) parameters are at the level of with the odd-parity coefficients (~ jk oþ) at the level of [7,8]. The disparity in these constraints arises because the sensitivity is determined by the symmetry of the sensing apparatus: the most sensitive cavity experiments are based on *Electronic address: fred@physics.uwa.edu.au Michelson-Morley-style experiments which are sensitive in leading-order to only the even ~ jk e parameters. Hence these parameters are the best constrained of the laboratorymeasured SME parameters. The sensitivity of even-parity experiments to odd-parity and isotropic SME coefficients arises solely because of the motion of the earth v relative to a sun-centered reference frame. For an even-parity experiment the sensitivity to the odd coefficients ~ jk oþ is reduced by the earth s velocity normalized to the speed of light: ¼ v c 10 4 )[6]. The sensitivity to the isotropic parameter ~ tr is reduced further by a factor of [9]. On the other hand, an odd-parity sensor can be leadingorder sensitive to the odd SME parameters ~ jk o [10,11], while only having first-order suppression of ~ tr see Table I. Hence an asymmetric odd-parity experiment can measure the odd-parity and isotropic SME parameters with enhanced sensitivity compared to previous even-parity Michelson-Morley type experiments. Here we report results from the first odd-parity optical resonator experiment and we are thus able to provide a constraint on the isotropic parameter ~ tr with the highest sensitivity yet reported. We further believe that this type of experiment has room for much improvement in the future whereas existing evenparity experiments are near the limit of development and are unlikely to improve by 4 orders of magnitude, limiting the potential for progress in the search for odd-parity and isotropic violations of Lorentz invariance. An analysis of an even-parity rotating microwave resonator experiment designed to test LI [12] has determined ~ tr as 15 7: [9]. An alternative means to determine ~ tr was obtained using relativistic ion spectroscopy [13] with sensitivity of 8: An oddparity interferometer has been used to determine ~ tr as 0: , limited by vibrational noise [14]. A number of other measurements of ~ tr have been performed based upon astrophysical observations [15], collider physics [16,17], or measurements of the electron spin [18] but =2011=84(8)=081101(5) Ó 2011 American Physical Society

181 FRED N. BAYNES, ANDRE N. LUITEN, AND MICHAEL E. TOBAR PHYSICAL REVIEW D 84, (R) (2011) TABLE I. Sensitivity to the SME parameters for different ðm DB Þ jk lab ¼ Re 1 sffiffiffiffiffiffi 0 Z types of resonators ( 10 ðe j 0c 2hUi Bk 0c ; 4 ). ÞdV 0 V Experiment Parameter Sensitivity Even-Parity ~ jk e ~ jk oþ 2 ~ tr where E and B are the components of the propagating Odd-Parity ~ jk oþ ~ tr ~ jk e electromagnetic fields and hui is the total energy in the mode. We see thus a secondary benefit of the use of counter-propagating modes; it makes the experiment twice these are indirect measurements or contain underlying as sensitive to a nonzero SME parameter when compared model assumptions [19]. with an experiment which uses the frequency shifts of a unidirectional beam in an asymmetric cavity. II. ASYMMETRIC OPTICAL RESONATOR The observable in a resonator-based test of LI is the normalized frequency shift = in the resonant frequency of the cavity. As an example, the best even-parity optical resonator experiment makes use of a rotating block of Ultra Low ExpansionÒ glass containing two symmetric orthogonal high finesse Fabry-Perot resonators in a heavily isolated and temperature controlled vacuum environment [7,8]. Any violation of Lorentz invariance will be manifested as modulations in the frequency difference between the two cavities, related to the rotation of the apparatus. Odd-parity experiments need to break the 180 rotational symmetry of an even-parity experiment. In the case of the experiment described here this asymmetry is achieved by placing a dielectric in one arm of a ring resonator, see Fig. 1. The requirement to include a dielectric element in the cavity means we cannot simply obtain temperature insensitivity by constructing the cavity from low thermal expansion materials such as Ultra Low ExpansionÒ glass. We overcome this drawback by sensing the frequency difference between counter-propagating modes, eliminating many of the causes of drift between the cavities. For example, most environmentally-driven changes in the optical path length are common to both counter-propagating modes and thus generate no effect on the frequency difference between the two modes. This makes the resonator insensitive to environmentally induced fluctuations, which is highlighted by the fact that no temperature control or vibration isolation was required. Using the derivation of resonator sensitivity to SME parameters outlined in [6], and those which specifically apply to an odd-parity cavity in [10,11], the only nonzero term contributing to the observable = is FIG. 1 (color online). Asymmetric ring resonator with UV fused silica prism at Brewster s angle B, showing counterpropagating modes. III. THE EXPERIMENT RAPID COMMUNICATIONS The asymmetric ring cavity was machined out of single aluminum block and is approximately 5cm 5cm, with one of the mirrors mounted between piezoelectric actuators for cavity length adjustment. The dielectric element is a UV fused silica Brewster angle prism (n ¼ 1:44) with a base of 1.7 cm. We are careful to use a Brewster s angle prism to minimize the surface losses together with a low dissipation dielectric material (UV grade fused silica). The measured finesse was 860 with contrast of 0.1 and a free spectral range of 3.85 GHz. Since the resolution of resonator experiments is set by the finesse of the cavity we used relatively high optical power to overcome the modest finesse and ensure optimal conditions for locking to the frequency of the two modes. Fluctuations associated with the high optical power (such as heating and nonlinear effects) are once again mitigated by the use of counterpropagating modes. We note that the low finesse was not a critical factor in this experiment as the frequency fluctuations in the relevant time domain are dominated by systematic fluctuations rather than limitations of the frequency locking. To monitor changes in the resonant frequency of the two counter-propagating modes the output of a laser is split into two paths which are independently frequency locked to the two fundamental modes counter-propagating in the optical resonator. The experiment has been designed to excite the fundamental mode of the optical resonator and we have experimentally verified that we are locked to the correct modes. This ensures that there is complete spatial overlap of the counter-propagating modes and rejection of optical path-length fluctuations. Higher-order modes are not frequency-degenerate with the fundamental mode and will not be excited while the laser is frequency-locked to the fundamental mode. We use the standard Pound-Drever- Hall (PDH) [20] technique to create the error signal required for frequency locking with the required phase modulation being provided by direct modulation of the laser crystal [21]. The use of two acousto-optic modulators (AOMs) allows independent frequency locks to each of the counter-propagating modes. The frequency is shifted by a constant 80 MHz ( 80 MHz ) in the first path using an AOM in the double-pass configuration [22]. This optical path then passes through a polarization-maintaining optical

182 RAPID COMMUNICATIONS TESTING LORENTZ INVARIANCE USING AN ODD-... PHYSICAL REVIEW D 84, (R) (2011) TABLE II. Sensitivity coefficients of ~ tr for this experiment (stationary) using the short data set approximation and differentiated data. Modulation Coefficient Numerical Value sinð! T Þ 2 cosðþ cosð 0 Þ 2: cosð 0 Þ ½ðM DB Þ XZ lab ðm DBÞ ZX lab Š cosð! T Þ 2 sinð 0 Þ 2: sinð 0 Þ ½ðM DB Þ XZ lab ðm DBÞ ZX lab Š FIG. 2 (color online). The optical setup, showing the use of two AOMs to eliminate the need for an optical beat note. fiber, half-wave plate and polarizer to ensure the correct polarization of light is incident on the cavity. The laser is locked to this resonance using the piezoelectric transducer on the laser. In the second path the locked laser light is sent through a second tunable AOM and this is used to provide the frequency corrections. The second AOM will thus have its frequency locked at 80 MHz plus any frequency difference between the counter-propagating modes ( mod ) see Fig. 2. By logging the frequency fluctuations of this second AOM we can measure the frequency difference between the two counter-propagating modes. The AOMs are powered by the amplified output of two signal generators. The signal generators and counters are all phase-locked to a common 10 MHz signal from a H-maser. Since any violations of LI will show only in the difference between the resonant frequencies of the two counter-propagating modes it is unnecessary to stabilize the cavity temperature to ensure a constant resonant frequency. However, in order to prevent large changes in operating conditions we use the piezoelectric transducers in the cavity and an additional slow loop to keep the laser at a relatively constant frequency. 0 and decomposes the signal for LI violations into sine and cosine terms with coefficients given in Table II. The coordinate system used for analysis in the SME is given in [6], with T the time since the laboratory frame y axis pointed towards 90 right ascension, the angle between the celestial orbital plane and the elliptic ( 23:40 ) and! is the sidereal frequency. In odd-parity experiments signals for LI violations occur at the rotation frequency (in the case of stationary experiment this corresponds to the sidereal frequency). The data set was divided into sections 2 days long and the amplitudes of the sinð! T Þ and IV. DATA ANALYSIS The data was acquired over 45 days from 19/11/2010 with about 35 days of usable data. Since the experiment is stationary in the laboratory we are searching for frequency modulations synchronous with earth s sidereal phase. To simplify and increase the speed of the analysis process we average the data into 20-minute blocks. The raw data is then differentiated with respect to sidereal phase to remove offsets and drifts [23]. The cavity is oriented in an East- West direction to maximize the sensitivity to possible violations of LI and this leads to sensitivity to only the cosð! T TÞ terms defined in Table I of [10]. As the data set comprises only a small section of the year we can apply the short data set approximation [24] which assumes a constant annual phase over the duration of the experiment FIG. 3 (color online). Difference in resonant frequency of counter-propagating modes. Processed time series data (top) and spectral density (bottom). The middle graph are the values obtained for ~ tr from the data set split up into 2 day portions

183 cosð! T Þ components are determined using a fitted least squares regression and from these two amplitudes a determination of ~ tr and ~ XZ oþ is made, and the quoted values are a weighted mean of all the data sets see Fig. 3. The value determined for the odd-parity parameter ~ XZ oþ is 0:7 1: which is an order of magnitude above the current limit. For the scalar parameter ~ tr the result is 3:4 6:2 10 9, a new limit on the constraint. The uncompetitive constraint placed on the odd parameter ~ XZ oþ compared to the scalar parameter ~ tr is because the current best constraints are derived from experiments with different sensitivities ([7,9], respectively) and in this first realization of an asymmetric optical resonator we use a nonrotating experiment, giving reduced sensitivity to the odd parameters [10]. V. DISCUSSION OF SYSTEMATICS Although common mode rejection of most environmental effects is a consequence of the counter-propagating mode design, there are nonetheless some systematic effects that afflict this experiment. In the PDH locking scheme unwanted residual amplitude modulation (RAM) copresent with the intended frequency modulation causes the laser to lock slightly off the center of resonance [20]. In usual PDH systems fluctuations in RAM will cause frequency fluctuations, although in our approach there is a rejection of this effect if the coupling and finesse of the counterpropagating modes were to be exactly the same. However, small alignment and mode-matching differences on the two modes leads to a small residual sensitivity to the level of RAM. The measured level of alignment fluctuations are consistent with the measured level of frequency fluctuations in this experiment when allowing for the mismatch of contrast on the two modes. Such systematic effects are the major source of instability in the experiment and are a limiting factor in the current constraint on ~ tr. There are nonreciprocal effects associated with the Faraday effect and stress birefringence in the fused silica that will cause a frequency difference between the two counter-propagating modes, related to the presence of magnetic fields in the laboratory [25]. Based on measurements of the magnetic field strength and variation near the optical resonator the calculated effect is 2 orders of magnitude below the uncertainty in ~ tr. The use of counter-propagating modes in a ring resonator means that the device will exhibit a sensitivity to rotational velocity in the plane of the device (Sagnac effect [26,27]). The Sagnac effect would affect the LI result only if the angular velocity or the dimensions of the optical resonator were to fluctuate with a sidereal FIG. 4 (color online). experiments. Comparison of ~ tr determined by other period but the presence of this systematic effect is more than 4 orders of magnitude below the uncertainty in ~ tr. These systematic effects are technical limits to the sensitivity of this particular experiment and can be drastically reduced through alignment and temperature control, magnetic shielding and a higher finesse or contrast cavity. VI. CONCLUSION An odd-parity experiment offers 10 4 times more sensitivity to the odd-parity ~ jk oþ and isotropic ~ tr parameters in experimental tests of LI. The value for ~ tr determined from this experiment is ~ tr ¼ 3:4 6: ð1errorþ, the tightest published constraint on ~ tr to our knowledge. This constraint is more than a factor of 12 better than previous test of LI at optical frequencies [13] and moderately better than the previous best [9] (see Fig. 4). This experiment is the first odd-parity optical resonator experiment and the use of counter-propagating modes enables a new constraint on ~ tr using a nonrotating resonator without temperature control, vibration isolation or vacuum systems. Given the inherent rejection of environmental fluctuations by the counter-propagating modes, sufficiently advanced oddparity experiments can now approach the sensitivity of the state-of-the-art Michelson-Morley type even-parity experiments. This would enable bounds on odd-parity ~ jk oþ and isotropic ~ tr to increase by up to 4 orders of magnitude, making odd-parity optical resonators an important experimental tool in the continuing search for violations of Lorentz invariance. ACKNOWLEDGMENTS RAPID COMMUNICATIONS FRED N. BAYNES, ANDRE N. LUITEN, AND MICHAEL E. TOBAR PHYSICAL REVIEW D 84, (R) (2011) This work was supported by the Australian Research Council Grants No. FL , FT and DP

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185 B Appendix High-performance Iodine Fiber Frequency Standard 169

186 4776 OPTICS LETTERS / Vol. 36, No. 24 / December 15, 2011 High-performance iodine fiber frequency standard Anna Lurie, 1, * Fred N. Baynes, 1 James D. Anstie, 1 Philip S. Light, 1 Fetah Benabid, 2 Thomas M. Stace, 3 and Andre N. Luiten 1 1 Frequency Standards and Metrology Group, School of Physics, The University of Western Australia, Perth, Western Australia Centre for Photonics and Photonic Materials, University of Bath, United Kingdom, Bath BA2 7AY, UK 3 Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia *Corresponding author: anna@physics.uwa.edu.au Received July 13, 2011; revised October 20, 2011; accepted November 4, 2011; posted November 9, 2011 (Doc. ID ); published December 14, 2011 We have constructed a compact and robust optical frequency standard based around iodine vapor loaded into the core of a hollow-core photonic crystal fiber (HC-PCF). A 532 nm laser was frequency locked to one hyperfine component of the R(56) I 2 transition using modulation transfer spectroscopy. The stabilized laser demonstrated a frequency stability of at 1 s, almost an order of magnitude better than previously reported for a laser stabilized to a gas-filled HC-PCF. This limit is set by the shot noise in the detection system. We present a discussion of the current limitations to the performance and a route to improve the performance by more than an order of magnitude Optical Society of America OCIS codes: , , The performance of optical frequency standards based on laser-cooled atoms and ions now surpasses the best microwave clocks [1,2]. However, there is a great deal of interest in developing simpler optical frequency standards for practical applications. One of the earliest optical frequency standards, based on iodine vapor, continues to inspire a great deal of work due to its simplicity [3 7]. Optical transitions of iodine continue to be included in the Bureau International des Poids et Measures recommended wavelengths for the practical definition of the meter [8,9], with recent work demonstrating fractional frequency stabilities of at 1 s[5 7] and at 1 year [7]. The approaches outlined in Refs. [5 7] are limited to the laboratory because they use large, fragile glass iodine cells. In contrast, an approach based on HC-PCFs could deliver the same long interaction region between gas and laser in a compact, robust, and portable format. Existing work on HC-PCF gas cells has largely focused on hydrogen-, acetylene- and rubidium-loaded fibers for spectroscopy [10 13]. Frequency standards based on gas-filled HC-PCFs were first demonstrated in 2005 [10]; to our knowledge the current state of the art is an acetylene-loaded HC-PCF with a fractional frequency stability of at 1 s[14]. We demonstrate an iodine-loaded HC-PCF as the core of a laser frequency stabilization system with a performance nearly an order of magnitude better than that earlier result. A simplified outline of the experimental system is shown in Fig. 1. A 1.3 m long Kagome structure HC-PCF with 25 μm core diameter [15] is mounted so that each end is close ( 1 mm) to a window inside a vacuum system, while the bulk of the HC-PCF sits outside the vacuum. A teflon fitting compresses a rubber cone onto the outside of the HC-PCF to create a vacuum seal around the fiber. The vacuum system includes a reservoir of solid iodine with a vapor pressure of 36 1 Pa (vapor pressure of iodine at room temperature 23.5 C) [16] in a separate valved section of the vacuum system. Counterpropagating pump and probe beams are coupled into opposite ends of the HC-PCF through 4 microscope objectives. The polarizations of the pump and probe are set orthogonal so that they can be easily separated at the output of the fiber. Because of unwanted reflections from the fiber, together with a small birefringence, residual pump light still falls on the photodiode that detects the transmitted probe signal. An acoustic optical modulator (AOM) shifts the pump beam by 200 MHz to reduce effects of this interference on the photodiode. This AOM is used to apply a frequency modulation to the pump beam to generate an error signal for the frequency locking system (see below). A 10 cm long traditional iodine gas cell at room temperature, with an independent pump probe beam pair, is used as a reference. We lock a 532 nm Innolight Nd:YAG laser to the a1 hyperfine component of the R56(32,0) manifold. This line exhibits a MHz linewidth (full width at halfmaximum) in the cell, and 15 2 MHz in the HC-PCF under operational conditions. The cell width arises principally from iodine iodine collisions, whereas the fiber width is due to iodine background gas collisions ( 3 MHz) and transit time broadening ( 3 MHz) [11,17] together with power broadening from the high intensities ( 9 MHz). Using the R(56) 32-0 on-resonance absorption we estimate the iodine pressure in the fiber as Pa, which is achieved after a 15 min loading period. Figure 2 shows the electronic locking system. The pump undergoes frequency modulation (FM) using the Fig. 1. (Color online) Simplified overview of the optical system, with counterpropagating pump (green) and probe (blue) beams. MO, microscope objective; AOM, acoustic optical modulator; PD, photodiode; HWP, half-wave plate; V, vacuum valves. The counterpropagating beams are separated for ease of visualization /11/ $15.00/ Optical Society of America

187 December 15, 2011 / Vol. 36, No. 24 / OPTICS LETTERS 4777 PZT LASER Synthesiser FM 4MHz (2MHz) deviation at 9.2kHz AOM Iodine saturated absorption spectroscopy Lock box FM lock-in Error signal AOM, which produces amplitude modulation on the probe according to the frequency difference between the laser and the hyperfine feature. This modulation transfer lock avoids issues associated with unwanted AM produced in conjunction with the FM. The peak FM deviation was chosen to produce suitable error signals (4 MHz for the fiber, 2 MHz for the cell). The bandwidth of this frequency control system is 150 Hz: the frequency at which the magnitude of the laser freerunning fluctuations equals the noise floor of the control system. High intensities in the fiber produce substantial Stark shifts of the iodine energy levels [18]. Fluctuations in the in-coupled power associated with vibration and mechanical drift of the alignment between the beams and fiber core cause flicker frequency fluctuations at the level of To overcome this we have incorporated active alignment for both the pump and the probe beams, which reduces the intensity fluctuations by a more than a factor of 5 at 1 Hz [19]. This additional stabilization reduced the power-driven fluctuations so that the standard now exhibited a white frequency limit consistent with detection shot noise. Figure 3 displays the fractional frequency stability of the free-running laser together with the performance when locked to a sub-doppler feature in both the fiber and cell. All measurements were made by generating a beat note against a high-performance cavity-stabilized laser [20] whose performance is also displayed in Fig. 3. The frequency stability is in terms of the conventional PD Fractional depth (%) Arbitrary units Freq (MHz) Hyperfine feature Freq (MHz) Fig. 2. (Color online) Simplified overview of the electronic locking system. Insets show the experimental data obtained at the error point and directly on the photodiode. Fig. 3. (Color online) Fractional frequency stability of the freerunning laser (red triangles), the laser locked to the 10 cm long iodine cell (blue crosses), the laser locked to the iodine-loaded HC-PCF (black squares), and the reference cavity (purple circles) against which these stabilities were measured. The limit to the frequency stabilization system for the fiber-stabilized laser (black line) is also displayed. square root Allan variance measure [21]. The fiber results shows a white frequency characteristic with a fractional frequency stability of at 1 s, which is almost an order of magnitude improvement over the best previous work using a gas-filled HC-PCF [14]. We have estimated the limits of the frequency control system by measuring the frequency error signal when the laser is tuned out of resonance with any iodine feature. Under these conditions the error signal will be sensitive to noise sources, such as any residual AM from the modulation locking scheme together with detector and shot noise; however, it will not be sensitive to effects that shift the iodine resonances (e.g., Stark, pressure and magnetic shifts). We see a white frequency noise floor, as shown in Fig. 3, which is consistent with the measured stability over 1 30 s integration range indicating that this detection noise is likely responsible for the performance in this range. We find further that this detection noise was associated with shot noise in the detection process with the measured current fluctuation at the output of the photodiode within 16% of the level calculated from first principles for the shot noise. For the cell-based iodine standard the input noise of the lock-in amplifier limited the performance a factor of 4 above the shot noise limit. Under operational conditions (see Table 1), the slope of the frequency discriminator in the fiber is about half that of the cell, due to the FM deviation used as well as three physical reasons: (1) the fractional depth of the Doppler-free features are 30 times higher in the fiber because of the high intensities and the excellent spatial overlap of the pump and probe beams; (2) the substantially lower fiber absorption (α a1 ) because the iodine pressure is 1 45th that of the cell; (3) the linewidth of the Doppler-free features in the fiber that are 3 times wider than that of the cell. This gives an expected fiber slope approximately equal to that of the cell, further reduced by a factor 2 3 because the FM deviation used for the fiber lock was half the optimum value. These observations show a route to substantially improving the short-term performance of the fiber frequency standard: one should have expected 45 times less collisional broadening in the fiber because of the lower iodine pressure; however, this was not seen Table 1. We List the Sub-Doppler Resonance Frequency Width, Measured Saturation Intensity, Operational Pump and Probe Intensities and Powers, the Doppler Broadened Absorption Coefficient at the Location of the a1 Hyperfine Component, and the Fractional Depth of the Hyperfine Component with Respect to the Doppler Broadened Absorption Experimental Parameters Cell Fiber Δ HF (MHz) I sat (kw m 2 ) I probe (kw m 2 ) I pump (kw m 2 ) P probe (μw) P pump (μw) α a α HF α a1 0.3% 9%

188 4778 OPTICS LETTERS / Vol. 36, No. 24 / December 15, 2011 because background gas (water/nitrogen/oxygen) is released during the iodine loading process. From the observed linewidth, we estimate a background pressure of 150 Pa [16], along with a 7% increase per hour in the hyperfine width due to continued outgassing. Refining the loading process to remove this background gas could reduce the low-intensity linewidth to that limited by transit time effects, which, due to the large mass of iodine, is 2 MHz at low pressures [11,17]: a value 3 times below that one would obtain for acetylene. A decreased rate of collisions would allow operation at lower pump powers, for a given sub-doppler signal strength, thereby alleviating Stark shift and power broadening effects. For maximum signal to noise on the discriminator one wishes the Rabi frequency of the pump to be of the same order as the intercollision frequency and under these conditions we calculate a possible reduction in linewidth from the current 15MHz to 7 MHz with the removal of the background gas. Furthermore, we currently do not have good control over the amount of iodine that can be loaded into the fiber. If we could reliably load an optimal density (αl 1), it would improve the discriminator signal by a factor of 9 from the results reported here. The combination of narrower linewidth, larger FM deviation and increased absorption depth could improve the frequency discriminator sensitivity by a factor of 30, allowing substantially improved short-term frequency stability. The compactness of the standard is likely to allow for effective temperature and magnetic shielding, although this will require the additional step of removing the fiber from the vacuum system [22]. Tight guidance of the pump and probe beams will minimize unwanted effects associated with fluctuations in their relative alignment. The combination of properties listed above suggest that long-term stability and reproducibility of the frequency standard can be good although this has not yet been directly tested. A laser has been locked to the a1 hyperfine component of the R56(32,0) transition in an iodine-loaded HC-PCF, with a fractional frequency stability of at 1 s limited by shot noise in the detection system. Future work will improve the loading scheme so that it is possible to load a specific quantity of iodine as well as avoiding contaminants entering the HC-PCF. The predicted performance of the frequency control system in this new standard should lie below at 1 s. We acknowledge the Australian Research Council for supporting this research. References 1. H. Katori, Nat. Photon. 5, 203 (2011). 2. H. S. Margolis, Contemp. Phys. 51, 37 (2010). 3. K. Nyholm, M. Merimaa, T. Ahola, and A. Lassila, IEEE Trans. Instrum. Meas. 52, 284 (2003). 4. G. D. Rovera, F. Ducos, J.-J. Zondy, O. Acef, J.-P. Wallerand, J. C. Knight, and P. St. J. Russell, Meas. Sci. Technol. 13, 918 (2002). 5. E. J. Zang, J. P. Cao, Y. Li, C. Y. Li, Y. K. Deng, and C. Q. Gao, IEEE Trans. Instrum. Meas. 56, 673 (2007). 6. J. Ye, L. Robertsson, S. Picard, L.-S. Ma, and J. L. Hall, IEEE Trans. Instrum. Meas. 48, 544 (1999). 7. J. Ye, L. S. Ma, and J. L. Hall, Phys. Rev. Lett. 87, (2001). 8. T. J. Quinn, Metrologia 40, 103 (2003). 9. R. Felder, Metrologia 42, 323 (2005). 10. F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, Nature 434, 488 (2005). 11. J. Hald, J. C. Petersen, and J. Henningsen, Phys. Rev. Lett. 98, (2007). 12. P. S. Light, F. Benabid, M. Maric, A. Luiten, and F. Couny, Opt. Lett. 32, 1323 (2007). 13. F. Benabid, P. J. Roberts, F. Couny, and P. S. Light, J. Eur. Opt. Soc. 4, (2009). 14. K. Knabe, S. Wu, J. Lim, K. A. Tillman, P. S. Light, F. Couny, N. Wheeler, R. Thapa, A. M. Jones, J. W. Nicholson, B. R. Washburn, F. Benabid, and K. L. Corwin, Opt. Express 17, (2009). 15. F. Couny, F. Benabid, and P. S. Light, Opt. Lett. 31, 3574 (2006). 16. T. Maisello, N. Vulpanovici, and J. W. Nibler, J. Chem. Education 80, 914 (2003). 17. C. J. Bordé, J. L. Hall, C. V. Kunasz, and D. G. Hummer, Phys. Rev. A 14, 236 (1976). 18. N. B. Delone and V. P. Krainov, Phys. Ispekhi 42, 669 (1999). 19. S. T. Dawkins and A. N. Luiten, Appl. Opt. 47, 1239 (2008). 20. J. J. McFerran, S. T. Dawkins, P. L. Stanwix, M. E. Tobar, and A. N. Luiten, Opt. Express 14, 4316 (2006). 21. D. W. Allan, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 34, 647 (1987). 22. P. S. Light, F. Couny, and F. Benabid, Opt. Lett. 31, 2538 (2006).

189 C Appendix Oscillating Test of the Isotropic Shift of the Speed of Light 173

190 PHYSICAL REVIEW LETTERS Oscillating Test of the Isotropic Shift of the Speed of Light Fred N. Baynes,* Michael E. Tobar, and Andre N. Luiten Frequency Standards and Metrology, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (Received 11 October 2011) In this Letter, we present an improved constraint on possible isotropic variations of the speed of light. 1 Within the framework of the standard model extension, we provide a limit on the isotropic, scalar parameter ~ tr of , an improvement by a factor of 6 over previous constraints. This was primarily achieved by modulating the orientation of the experimental apparatus with respect to the velocity of Earth. This orientation modulation shifts the signal for Lorentz invariance to higher frequencies, and we have taken advantage of the higher stability of the resonator at shorter time scales, together with better rejection of systematic effects, to provide a new constraint. 2 DOI: PACS numbers: Ft, p, Cp, Da Introduction. Experimental tests of Lorentz invariance (LI) have played a prominent role in the progress of physics over the last 120 years. Beginning with failed searches for the luminiferous aether wind and moving to the confirmation of special relativity, the results of speed of light experiments provide essential confirmation for theories at the foundation of modern physics. A renaissance of interest in tests of LI has been driven by the postulates of new theories that violate LI while attempting to unify general relativity and the standard model [1 4]. Resonator experiments search for violations of LI through position, velocity, and orientation dependencies of the speed of light. A theoretical framework is required to report and interpret the results of these experiments. An early example was the Robertson-Mansouri-Sexl framework [5,6], where limits are set on parameters which represent deviations from the expected results of special relativity. In this framework the experiment is analyzed with respect to the reference frame of the cosmic micro- 5 wave background. Recently a more comprehensive framework the standard-model extension (SME) [7,8] has been widely used. The SME is a parameterization of all possible violations of LI by known fields and is analyzed with respect to a sun-centered celestial equatorial frame (SCCEF) moving inertially with respect to the cosmic microwave background. The SME has only recently been extended from the minimal SME to include operators of higher dimension [9,10]. In these theories, the choice of reference frame is arbitrary; however, any nonzero parameter represents a violation of LI. In the photon sector of the minimal SME, there are 19 independent parameters representing violations of LI through various deviations in the universal speed of light. The parameters of the photon sector of the SME are accessed through astrophysical observations and terrestrial resonator experiments with vastly differing sensitivities based on the properties of the relevant parameter and the experiment. The constraints on the 10 parameters representing vacuum birefringence (~ jk eþ and ~ jk o ) are based on astrophysical observations of pulsed sources and broadband polarized sources [11]. The vast distances involved with astrophysical observation have allowed tight constraints to be placed on the birefringent parameters at the level of 10 32, and these parameters are generally set to zero in calculations for the remaining 9 parameters. The remaining SME parameters together create a timeindependent but orientation-dependent modification to the speed of light parameterized in the SCCEF. Essentially, the isotropic ~ tr is an average shift over all possible directions: ~ jk e represents a directional dependence in the speed of light and ~ jk oþ is the relative difference between light moving parallel and antiparallel to some particular direction. Laboratory-based experiments are sensitive to combinations of these SME parameters through Lorentz transforms from the SCCEF to the laboratory reference frame, and a time dependence on the parameters is induced through the relative orientation of the Earth. Advanced, modern versions of Michelson-Morley (MM) experiments have constrained the even-parity paramter ~ jk e terms to the level and the odd-parity ~ jk oþ terms to level [12,13]. The even-parity symmetry of the the MM experiments gives rise to first-order sensitivity to the even-parity terms and a reduced sensitivity to the oddparity terms. The reduction in sensitivity to the odd-parity ~ jk oþ terms depends on the velocity of the experiment, and the disparity in sensitivity is given by the velocity of the Earth normalized to the speed of light: ¼ v c 10 4 [11]. The isotropic SME parameter ~ tr suffers a further 2 ( 10 8 ) reduction in sensitivity from even-parity MM experiments [14]. However an odd-parity resonator has only a first order,, reduction in sensitivity to ~ tr [15] and provides an avenue to improve the constraints on the frame-dependent isotropic shift of the speed of light. Tests of the isotropic shift of the speed of light. The SME parameter ~ tr can be derived from numerous types of experiments. The absence of photon decay events from 1 Ó 2012 American Physical Society 1

191 PHYSICAL REVIEW LETTERS high-energy cosmic rays yields a constraint at the level of [16], based on assumptions of the high-energy dynamics. Further constraints are derived from particle accelerators, the lack of vacuum Cherenkov radiation from relativistic electrons [17,18] atthe10 11 level, and the characterization of synchrotron emission rates at the level [18]. Contributions of ~ tr to the anomalous magnetic moment of the electron can be calculated and compared to the standard value, giving a constraint at the 10 8 level. However, the experiments mentioned above contain model dependent assumptions or measure ~ tr indirectly [19]. Direct measurements of ~ tr can be derived from experiments using spectroscopy on fast-moving ions to measure the Robertson-Mansouri-Sexl time-dilation parameter [20,21]. The SME parameter ~ tr can be obtained from these experiments by considering the phase velocity of signals traveling in opposite directions in the laboratory frame moving with respect to a sun-centered reference frame which provides a constraint at the 10 8 level [15]. Resonator tests of LI yield the tightest constraints for direct measurements of ~ tr. Rotating, even-parity microwave resonators have determined ~ tr as 15 7: [14] and a rotating odd-parity microwave resonator reached the 10 7 level, limited by vibrational noise [22]. The best constraint yet reported is by an odd-parity optical resonator with ~ tr ¼ 3:4 6:7 10 9, from an experiment stationary in the laboratory [23] using counterpropagating modes. The experiment reported here is an improved version of [23], utilizing orientation modulation together with alignment control to reduce systematic errors and improve the constraint on the isotropic shift of the speed of light by a factor of 6. Asymmetric optical resonator and optical setup. A violation of LI in resonator experiments manifests itself as a shift in the resonant frequency of an optical cavity dependent on the orientation with respect to the SCCEF. In this experiment we use the frequency difference of counter-propagating modes as the observable, and the orientation of the cavity in the laboratory frame is rotated 180 approximately every 10 minutes. The resonant cavity is an odd-parity asymmetric ring resonator with a dielectric in one arm of the ring to provide the necessary asymmetry. The dielectric is a Brewster s angled UV-fused silica prism (n ¼ 1:44) with an optical path length through the prism of 14 mm. The cavity has a finesse of 860 and a free spectral range of 3.87 GHz and is housed in a single machined aluminum block with high-reflectivity dielectric mirrors. The cavity is designed so that the optical path strikes the Brewster s angled prism at the correct angle to minimize losses for p-polarized light and the lack of orthogonal surfaces inhibits the generation of reflected modes; see Fig. 1. There are ring piezo-electric transducers on one of the mirrors to enable adjustments of the cavity length to ensure continuous long term operation of the experiment. FIG. 1 (color online). Diagram of the experiment showing the asymmetric ring resonator; there is an equivalent setup for the counterpropagating mode. 11 A 1064 nm diode-pumped, nonplanar ring laser is split into two paths that are independently frequency locked to counter-propagating modes using the standard Pound- Drever-Hall technique [24] with a 987 khz modulation frequency. The observable in the experiment is the frequency difference of counter-propagating resonant modes. The two incoming beams are mode matched to the fundamental mode of the resonator ensuring complete overlap of the spatial mode of the resonator and we verify that we are exciting the fundamental transverse mode. The use of counterpropagating modes allows the rejection of systematic cavity length fluctuations driven by environmental fluctuations. A change in the optical path length is common to both directions of propagation and will be rejected in the frequency difference signal. This is a major advantage of the asymmetric ring resonator and the experiment reported here could be performed without temperature control or vibration isolation with no loss of performance. To measure the frequency difference between the counterpropagating modes, the laser is frequency locked to the resonator by two independently actuated acousto-optic modulators (AOMs) in the double pass configuration [25], and the difference frequency is monitored by a frequency counter. This is achieved by frequency shifting one of the incoming beams by an AOM by a fixed 160 MHz and then controlling the frequency of the laser to lock to the propagating mode of the resonator. The second beam passes through an independent AOM to lock to the counterpropagating mode, and any frequency corrections are provided by the second AOM [23]. Thus the correction signal sent to the second AOM represents the frequency difference between the two counterpropagating modes, and in this signal we search for violations of LI. Spatial modulation and alignment control. By rotating the asymmetric resonant cavity by 180 in the laboratory, the sign of expected signal for violations of LI, caused by nonzero ~ tr, is reversed. This is due to the change in orientation of the cavity with respect to the velocity of the Earth, in a similar manner that the rotation of the earth gives rise to expected signals with sidereal period. The orientation modulation of the cavity is analogous to the optical chopping technique widely used in experiments to 2 2