Upper Limit to the Transverse to Longitudinal Motion Coupling of a Waveguide Mirror
S. Leavey, B. W. Barr, A. S. Bell, E-B. Kley, N. Gordon, C. Gräf, S. Hild, S. H. Huttner, S. Kroker, J. Macarthur, C. Messenger, M. Pitkin, B. Sorazu, K. Strain, A. Tünnermann
UUpper Limit to the Transverse to LongitudinalMotion Coupling of a Waveguide Mirror
S Leavey B W Barr A S Bell N Gordon C Gr¨af S Hild S H Huttner E-B Kley S Kroker J Macarthur C Messenger M Pitkin B Sorazu K Strain A T¨unnermann Correspondence: [email protected]
Abstract
Waveguide mirrors possess nano-structured surfaces which can poten-tially provide a significant reduction in thermal noise over conventional di-electric mirrors. To avoid introducing additional phase noise from motionof the mirror transverse to the reflected light, however, they must possessa mechanism to suppress the phase effects associated with the incidentlight translating across the nano-structured surface. It has been shownthat with carefully chosen parameters this additional phase noise can besuppressed. We present an experimental measurement of the coupling oftransverse to longitudinal displacements in such a waveguide mirror de-signed for 1064 nm light. We place an upper limit on the level of measuredtransverse to longitudinal coupling of one part in seventeen thousand with95% confidence, representing a significant improvement over a previouslymeasured grating mirror.
1. SUPA, School of Physics and Astronomy, The University of Glasgow, Glas-gow, G12 8QQ, UK2. Friedrich-Schiller-University, Abbe Center of Photonics, Institute of AppliedPhysics, Max-Wien-Platz 1, 07743 Jena, Germany3. Fraunhofer Institute of Applied Optics and Precision Engineering, Albert-Einstein-Str. 7, 07745 Jena, Germany
Major upgrades to the worldwide network of gravitational wave detectors arecurrently under way. New designs for the Advanced LIGO [1], Advanced Virgo[2], KAGRA [3] and GEO-HF [4] detectors will provide unmatched ability todetect gravitational waves in the audio spectrum. At their most sensitive fre-quencies, these detectors are expected to be limited by Brownian thermal noisearising from the reflective coatings on the detectors’ test masses [5–8]. In order1 a r X i v : . [ a s t r o - ph . I M ] A ug o help mitigate this limitation beyond the next generation of detectors, effortsare under way to develop mirror coatings with lower thermal noise [9, 10].In the case of Advanced LIGO, each end test mass (ETM) consists of asubstrate with 19 pairs of sub-wavelength coatings which produce a transmissionof 5 ppm for 1064 nm light [11]. Each layer within this stack contributes tothe overall thermal noise [7, 8]. The approach taken by Levin to calculate thethermal noise of mirrors [5] shows that mechanical loss at the front surface of amirror contributes more to the Brownian noise level than loss from an equivalentvolume in the substrate. Additionally, typical coating materials tend to exhibitmechanical loss orders of magnitude higher than typical substrate materials[7, 8]. For these reasons particular attention is being given to the reduction ofcoating thermal noise to improve the sensitivity of future detectors.One strategy, to be applied for example in KAGRA, is to cool the mirrorsto cryogenic temperatures. While this can potentially reduce the thermal noiseof the mirrors [12], the application of cryogenic mirrors requires new infrastruc-ture, different choices of mirror substrate and coating materials and poses thechallenge of heat extraction from the mirror without spoiling its seismic isolationand thermal noise performance. Efforts in the application of cryogenics are alsounder way to identify suitable substrate and coating materials for ET-LF, thelow frequency interferometer as part of the proposed Einstein Telescope [13–16].Apart from using different coating materials [17, 18] or different beam shapes[19–21] such as with LG33 modes [22], another potential approach is to utilisewaveguide mirrors (WGMs) [23–26]. These mirrors can possess high reflectiv-ity at a wavelength determined by their structure. In contrast to conventionaldielectric mirrors, mirrors possessing waveguide coatings can exhibit high re-flectivity without requiring multiple stacks [27]. A waveguide coating insteadpresents incident light with a periodic grating structure of high refractive indexmaterial n H on top of a substrate with low refractive index n L (see Figure 1).Light is forced into a single reflective diffraction order, the 0 th . In transmission,only the 0 th and 1 st diffraction orders are allowed as long as the condition inEquation 1 for the grating period, p ; and the light’s wavelength in vacuum, λ ,is fulfilled [23]. The light diffracted into the 1 st order undergoes total internalreflection at the substrate boundary where it excites resonant waveguide modes.Light leaving the waveguide then contains a 180 ◦ phase shift with respect tothe 0 th order transmitted light, causing destructive interference such that mostof the incident light is reflected [28]. λn H < p < λn L (1)A recent set of calculations by Heinert et al. [29] showed that a suitablyoptimised WGM can provide a reduction in coating thermal noise amplitudeof a factor of 10 at cryogenic temperature compared to mirrors employed inAdvanced LIGO.Previous efforts to demonstrate grating structures as alternatives to dielec-tric mirrors have identified phase noise in the light reflected from the grating2 aveguide (n H )substrate (n L )destructive interferenceconstructive interference grating structure Figure 1: Propagation of light within a waveguide mirror. The grating andwaveguide layers have refractive index n H , and sit atop a substrate of refractiveindex n L . Blue arrows represent incident light and red arrows represent reflectedlight. In realisations of waveguide mirrors such as this, a thin etch-stop layer isplaced between the grating and waveguide layers to assist fabrication [26].not otherwise present in dielectric mirrors [30, 31]. This effect arises from trans-verse motion of grating mirrors with respect to the incident light. Incident lightat angle α is reflected into the m th diffraction order, exiting at angle β m (seeFigure 2). The change in path length δl L between the reflected and incidentlight is then δl L = ζ a + ζ b = δy (sin α + sin β m ) , (2)where ζ a and ζ b represent the relative optical path length of each depicted ray.The phase modulation induced in the light reflected from the WGM is propor-tional to Fourier frequency with a 90 ◦ phase lead over the transverse motion [32].The noise added to the reflected light can be enough to mitigate the improve-ment in coating thermal noise, as witnessed in a study of 2 nd order Littrowgratings [32]. Although WGMs also possess gratings, the resonant waveguidestructure has been shown in simulations by Brown et al. to be invariant totransverse to longitudinal coupling [33].There are two mechanisms by which grating mirrors can couple transversemotion into longitudinal phase changes (see Figure 3). The first is throughtransverse motion of the grating, which can in principle be minimised with ap-propriate suspension design. The second mechanism is the coupling of changesin the opposite cavity mirror’s alignment into the spot position on the gratingmirror. This effect is of particular importance to gravitational wave observa-tories, where longer arm lengths can increase its detrimental impact. For thisreason the second mechanism is considered in more detail in this work.In order to quantify its transverse coupling, a WGM was produced in col-laboration with Friedrich-Schiller University Jena, Germany (see Table 1 forits properties). It was designed for light of wavelength 1064 nm, and consistedof an etched grating structure on top of a waveguide layer, both tantala, on a3 𝛂 𝛃 m 𝛇 a 𝛇 b δ y y Figure 2: Optical path length changes ζ a and ζ b due to transverse motion of aLittrow grating. Incident light diffracted into a different order undergoes a pathlength change δl L = ζ a + ζ b . Parameter Value
Materials SiO , Ta O ,Al O Design λ The fabricated WGM was used as the input coupler for a Fabry-P´erot cavity,held on resonance using the Pound-Drever-Hall (PDH) technique [34]. The errorsignal provided by the PDH technique represents changes in cavity length, andthis can be fed back to the laser’s frequency via a frequency stabilisation servo.
A non-zero WGM transverse to longitudinal coupling, ω , produces a phaseshift on the reflected light. This manifests itself as an effective change in cavitylength, δl W , as the laser light is scanned across its grooves by a rotation of theETM: δl W ( θ, κ, ω ) = θκω , (3)where θ is the ETM’s rotation angle and κ is the cavity’s coefficient of ETMrotation to transverse WGM spot motion.Additional cavity length changes are also produced via two geometrical ef-fects (see Figure 4). The first effect, δl s , is due to the position of the beam5 dy y d Figure 4: Geometrical ETM longitudinal effects. For a given rotation θ and spotcentre position offset y , the (longitudinal) position change in the surface of themirror (show in blue) as seen by the reflected light is approximately yθ + d θ .The straight, solid red line in the figure shows this longitudinal change.with respect to the centre of the mirror’s surface. For a rotation θ , a beamoffset from the centre of the mirror by a displacement y will receive a change in(longitudinal) path length of δl s ( y, θ ) = y tan θ ≈ yθ (4)for small angles. The second effect, δl d , is due to the depth d of the mirror,proportional to the rotation angle θ . The position of the centre of the mirrorwith respect to the zero rotation case, y d , is then y d ( d, θ ) = d θ ≈ d θ, (5)and the change in path length this causes is δl d ( d, θ ) = y d tan θ ≈ d θ . (6)The total longitudinal effect δl E caused by the rotation of the ETM is therefore δl E ( y, θ, d ) = δl s + δl d ≈ yθ + d θ . (7)Considering the ETM’s level of rotation and its dimensions and mass, it ispossible to calculate the cavity length change due to the two geometrical effects6 -4 -3 -2 -1 C a v i t y l e n g t h c h a n g e [ m / r a d ] ETM RotationTotal 1Total 2Total 3WGM 1WGM 2WGM 3 P h a s e [ ° ] ETM RotationTotal 1Total 2Total 3
Figure 5: Simulations of indicative cavity longitudinal error signals during ETMrotation for different levels of WGM coupling. The signals are functions ofthe transverse position of the reflected light relative to the ETM’s centre ofrotation, the angle of rotation, the mirror depth and the WGM’s coupling level.The rotation to longitudinal coupling of the ETM (black dashed line) combineswith the transverse to longitudinal coupling of the WGM (red, green and bluedashed lines) to produce cavity length changes (red, green and blue solid lines).In this example configuration, the ETM rotation is 1 × − rad, the ETM’sdepth is 0.1 m and the corresponding WGM coupling levels are 1:370 (red),1:3700 (green) and 1:37000 (blue).shown in Equation 7 and then, from the residual cavity length change, infer theWGM’s coupling level. The phase effect associated with transverse to longitu-dinal coupling is expected to be independent of spot position, whereas there is aphase change about the ETM’s centre of rotation. It is therefore expected thata spot position will exist, for a non-zero WGM transverse coupling level, offsetfrom the ETM’s centre of rotation, for which there is a cavity error signal min-imum. This effect arises as a result of δl W and δl E combining coherently (seeFigure 5). The spot position corresponding to the cavity error signal minimumallows the WGM’s transverse to longitudinal coupling level to be inferred.Examples of WGM coupling levels yielding cavity length changes smallerthan (blue), larger than (red) and roughly equivalent to (green) the ETM’seffects are shown in Figure 5. For cases where the WGM’s coupling level yieldsa significant cavity length change with respect to that of the ETM’s rotation,coherent combination creates a trough offset from the ETM’s centre of rotation.7 .2 The Glasgow 10 m Prototype The Glasgow 10 m prototype facility provided a test bed in which the WGM’stransverse to longitudinal coupling could be quantified. The prototype is housedin a Class 1000 clean room and consists of an input bench at atmospheric pres-sure and a vacuum envelope able to reach pressures of order 10 − mBar. Theenvelope consists of nine 1 m diameter steel tanks, each connected by steel tubes,arranged into two parallel arms of length 10 m, with a shorter arm for input op-tics situated between them.In the experiment, 1064 nm laser light was passed through a single-mode fibreto provide spatial filtering and an electro-optic modulator (EOM) to impose RFsidebands on the light to facilitate PDH control. The light was then coupledinto the vacuum system via a periscope. This configuration can be viewed inFigure 6.Tanks 2 and 3 housed a beam splitter and steering mirror, respectively,attached to double stage suspensions. In tanks 4 and 5 were sets of two triplesuspension chains based on the GEO-600 design [35]. A viewport present to therear of tank 5, and to the side of tank 1, allowed for light to exit the vacuumenvelope for the purposes of sensing and control.The WGM was attached to an aluminium block of mass 2.7 kg and suspendedfrom tank 4’s cascaded (triple) pendulum, forming the cavity’s ITM. A silicatest mass, also 2.7 kg, with a 40 ppm transmission coating, was used as theETM, suspended from a similar triple pendulum in tank 5. On the rear surfaceof the ETM were three magnets for the purpose of actuation, the positions ofwhich are shown in Figure 7. With optimal alignment the mirrors formed anovercoupled cavity with finesse 155.A three-stage reaction chain was placed behind the triple pendulum of theETM to provide voice coil actuation upon the magnets on the ETM’s rear sur-face. The upper and intermediate stages were identical to those of the chaincarrying the ETM, however—for the purposes of another experiment, not re-ported here—the lower stage was split into two parts separately suspended fromthe intermediate stage. The part closer to the ETM was a 1.8 kg aluminiumblock that carried the voice coils. The other part was a 0.9 kg aluminium blockrequired to balance the suspension. An RF photodetector was placed at the viewport on tank 1, where it could viewthe light reflected from the cavity. By using PDH demodulation, the signal fromthis photodetector provided an error signal for the cavity length. This signalwas fed back to the laser via the frequency stabilisation servo to maintain cavityresonance. The frequency stabilisation servo’s high frequency feedback signal—avoltage applied across the laser’s piezoelectric transducer (PZT)—provided ameans of calibrating cavity length changes at frequencies greater than 12 Hz.Using the PZT’s frequency response, 1.35 MHz/V rms , the cavity length change δl per error signal volt could be calculated to be 133 nm/V peak .8 ank 2Tank 1 Tank 3Tank 4Tank 510m Fabry-PerotcavityRF Photodiode CCDcameraEOM10 MHzsourceMixer 70 HzsourcesFrequencystabilisationservo Modecleaning fi breFast feedback(PZT)Slow feedback(temperature) 1064 nmlaserDataacquisitionsystem CoilsMagnetsETMReactionmass
From 70 Hzsources
Figure 6: The experimental setup in the prototype facility. The laser light ispassed through input optics (not shown), a mode cleaning fibre and an EOMbefore being coupled into the vacuum system via a periscope. It then travelsto tank 2 where it is reflected off a beam splitter and directed into one of thearms of the prototype by a steering mirror in tank 3. The two cavity mirrorsin tanks 4 and 5 form a Fabry-P´erot cavity. The cavity mirrors are suspendedfrom triple stage suspensions, and the beam splitter and steering mirror areboth suspended from double suspensions.The ETM is rotated in yaw using the 70 Hz source. It is fed to a coildriver where it is coupled into tank 5 via a vacuum feedthrough. Coil formerson the front edges of the reaction mass contain wound copper wire connectedto the vacuum feedthrough. Magnets are attached to the back of the ETM.The reaction mass is behind the ETM, containing a hole in its centre to allowlight to exit the vacuum tank where it can be viewed with the CCD camera. Alarger version of the contents of tank 5 can be viewed in the panel to the rightof the figure.The cavity is held on resonance by the frequency stabilisation servo. This feedsback to the light’s frequency via the laser crystal’s temperature below 12 Hzand its PZT above 12 Hz up to a unity gain frequency of 14 kHz.9
Figure 7: The positions of the magnets on the rear surface of the ETM. Themagnet designations used in this article are shown in red text. The top magnetis positioned at the centre of yaw, near the top of the mass. The left and rightmagnets are positioned 56.3 mm either side of the centre of yaw. Coils on theETM’s reaction mass (not shown) are positioned coaxially behind each magnet.
Parameter Description
Cavity input power Approx. 150 mWETM transmissivity 40 ppmETM radius of curvature 15 mETM spot size 2.138 mmITM transmissivity 4 %ITM radius of curvature ∞ ITM spot size 1.554 mmCavity length 9.81 mCavity finesse 155Cavity g-factor 0.347Beam waist size 1.554 mmBeam waist position At ITMSideband frequency 10 MHzTable 2: Cavity parameters.10
Measurements and Analysis
From the orientation of the WGM’s gratings, it was expected that actuation ofthe ETM in yaw, which would scan the cavity light across the WGM’s surfacetransverse to the direction of its grooves, would exhibit WGM transverse tolongitudinal coupling if present.For the purposes of actuation upon the ETM, two sinusoidal signals V L and V R (corresponding to the left and right voice coils on the ETM’s reaction mass,respectively) were produced using separate, phase locked signal generators. Asignal frequency of 70 Hz was chosen so as to be above the suspensions’ polefrequencies but low enough to provide an adequate signal-to-noise ratio. Thesignals V L and V R , with suitable balancing (see below), could then be actuatedin- or out-of-phase to produce longitudinal or yaw actuation upon the ETM,respectively.When V L and V R were identical in magnitude but out-of-phase, the ETM’smovement contained a linear combination of rotational and longitudinal compo-nents due to force imbalances between the voice coils. To ensure that actuationupon the ETM contained only a yaw component, the cavity’s longitudinal er-ror signal was minimised during out-of-phase actuation by changing the gain of V L . This balanced the magnitude of the torque applied by each actuator to theleft and right sides of the ETM. Any WGM transverse to longitudinal couplingpresent would act with phase orthogonal to this voice coil actuation and wouldthus be unchanged by the torque balancing.Pitch actuation upon the ETM, which would scan the cavity light in a di-rection parallel to the WGM’s grooves, was not expected to contribute to thecavity’s error signal via the WGM’s coupling. However, unintended pitch actu-ation upon the ETM would couple into the cavity’s length via the same geo-metrical mechanism as yaw shown in Equation 7. To minimise the ETM’s pitchcomponent during actuation in yaw, the cavity’s error signal was minimised byapplying an offset voltage to the top coil. In practice, minimal pitch couplingwas achieved when the offset signal was zero. To calibrate the cavity’s longitudinal response to voice coil actuation, the voicecoils were actuated with the balanced V L and V R signals in-phase at a frequency f = 70 Hz for a period of 120 s. This, along with the ETM’s mass m , could thenbe used to obtain the force applied to the ETM by the voice coils: F = 4 π f mδl. (8) Four spot positions corresponding to y in Equation 4 were chosen across thesurface of the ETM. The input beam was aligned to the cavity axis correspond-11 pot position [ mm ] − − − − − − − − V L . All spot positions have an error of +/-1 mm.ing to each spot position using the beam splitter and steering mirror nearestto the ITM, and the cavity mirrors were aligned to create a fundamental moderesonance. The voice coil signals V L and V R were set out-of-phase to producemotion on the ETM in yaw. The magnitudes of V L and V R were not alteredbetween the longitudinal calibration and this yaw actuation, so it was expectedthat the previously outlined minimisation of yaw to tilt actuation would alsoresult in minimal longitudinal to tilt actuation. The cavity length signal wasrecorded for a period of 300 s.For each nominal spot position an additional measurement was taken with V L set to ± . via the CCD camera placed intransmission of the ETM, using the known width of the ETM’s reaction massas a calibration. The error in the spot position measurements dominated theerror in voice coil alignment. Although misaligned voice coils could have lead toa change in the expected ETM force coupling (leading to a change in the centreof rotation of the ETM), it was found from separate measurements that theeffect of any possible misalignment during the experiment could only accountfor a drop in force of 0.11 %. This contributed a negligible error (+/-0.03 mm)to the results.Knowledge of the distance of the ETM’s voice coils from the centre of ro-tation, y c ; the ETM’s moment of inertia, I ; the coil driving frequency, f ; andthe force calibration from Equation 8, allowed the rotation angle to be obtained12eometrically using the relation θ = F y c π f I . (9)The numerical simulation tool
FINESSE [36] was then used to calculate κ for thecavity parameters shown in Table 2. This was determined to be 18.5 m rad − .The WGM’s transverse displacement was then the product of κ and θ . Using the known contribution to the cavity length signal from the rotation ofthe ETM, δl E , and the cavity length signals δl measured during the experiment,the WGM’s coupling level could be calculated statistically using Bayes’ theorem.For this experiment, Bayes’ theorem can be expressed mathematically as: p ( (cid:126)ω |D ) ∝ p ( D| (cid:126)ω ) p ( (cid:126)ω ) , (10)where p ( (cid:126)ω |D ) is the probability density distribution of the experimental param-eters, (cid:126)ω , given the observed data, D (the posterior ); p ( D| (cid:126)ω ) is the likelihoodand p ( (cid:126)ω ) is the probability distribution of the experimental parameters. Theobserved data D are the measured cavity error signals for each of the spotpositions.In this analysis we are primarily interested in estimates of the model param-eters. We are therefore free to ignore the constant evidence factor p ( D ) presentin Bayes’ theorem when calculating the posterior. In the future it may be ofinterest to compare different models for the coupling level (or lack thereof), inwhich case the evidence could be calculated to obtain a model odds ratio. To obtain a posterior for the WGM’s coupling level, it was necessary to build amodel and state prior belief of the parameters’ probability distributions.In the model, the ETM’s geometrical longitudinal effect at arbitrary spotposition y (Equation 7) for the rotation and mirror depth used in the exper-iment was combined coherently with a specified level of WGM transverse tolongitudinal coupling, ω . It was then possible to predict the total change incavity length δl as a function of spot position y , given the fixed parameters θ , κ and d , using equations 3 and 7: δl ( (cid:126)ω, y, θ, κ, d ) = δl W ( θ, κ, ω ) + δl E ( y, θ, d ) ≈ θκω + yθ + d θ . (11)The effect of beam smearing was also considered. The suspended optics con-tain residual displacement noise, leading to a broadening of the trough at whichthe ETM’s longitudinal coupling and any WGM coupling cancel (see Figure 5).To model this effect, the assumption was made that the motion of the spots13n the ETM followed a Gaussian distribution about their nominally measuredposition. Eight-hundred small ‘offset distances’ δy were applied uniformly tothe spot positions, drawn from a randomly generated Gaussian distribution.The number of offset distances was chosen as a compromise between adequatestatistical significance and technical constraints. Calculating the cavity lengthchange as a function of spot position for each of these offset positions, and com-bining them in an uncorrelated sum, allowed an average, ‘smeared’ signal to bemodelled which more closely resembled the measurements. The standard devi-ation of the Gaussian distribution was an additional parameter, ω , provided asan input to the model.The summing of signals introduced by the modelling of beam smearing ledto an artificial increase in the magnitude of the model’s predicted cavity lengthsignals. To compensate for this effect, a further parameter was introduced: amultiplicative scaling factor, ω , applied uniformly to the model. This factor alsohad the additional effect of compensating for the uncertainty in the calibratedcavity length signals. By marginalising over a suitable distribution of scalingfactors, it was possible to account for this uncertainty in the analysis of theWGM’s coupling level. The model used in the analysis to predict the smeared,scaled cavity length change, δl (cid:48) , was then: δl (cid:48) ( (cid:126)ω, y, θ, κ, d ) = ω (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) i =1 δl ( (cid:126)ω, y + δy i , θ, κ, d ) , (12)where δy i is the i th offset distance, drawn from a Gaussian distribution withstandard deviation ω . The likelihood function assumed for the model was a Gaussian distribution, p ( (cid:126)ω |D ) ∝ exp (cid:32) − N (cid:88) i =1 ( D i − δl (cid:48) ( (cid:126)ω, y i , θ, κ, d )) σ (cid:33) , (13)where N is the number of spot positions and σ is the (identical) variance ofeach of the measured spot positions. Bayes’ theorem requires an assumption of probability distributions ( priors ) foreach of the free parameters prior to the consideration of the measured data.The assumptions made for each free parameter in the model can be found inTable 4. The upper bound on coupling was assumed to be a factor 10 betterthan the grating mirror measured in [32], given the indication from [33] thatno coupling is present. The bounds on the scaling factor and spot smearingstandard deviation were chosen from earlier observations of the behaviour ofthe signals during the experiment. All priors were assumed to be uniform.14 arameter Symbol Distribution Dimensions
WGM transverseto longitudinalcoupling ω Uniform, (cid:2) , (cid:3) m (longitudinal)m (transverse) Spot smearingnoise standarddeviation ω Uniform, (cid:2) , × − (cid:3) m (transverse)Calibration scaling ω Uniform, (cid:2) , (cid:3) Table 4: The distributions assumed for each of the free parameters in the model,along with their dimensions, prior to the computation of the posterior.
A form of the Metropolis-Hastings Markov-Chain Monte-Carlo (MCMC) algo-rithm [37] was applied to the model to marginalise over the three parameters.The outputs of the MCMC are a chain of samples (values at each parameter)that are drawn from the posterior distribution. A histogram of samples for agiven parameter gives the marginal posterior distribution for that parameterfrom which the mean and standard deviation can be calculated.To ensure the convergence of the MCMC on the posterior, a ‘burn-in’ periodof 100 000 iterations was performed. The convergence was verified manuallyfollowing completion. A further 100 000 iterations were then used to samplefrom the posterior and this second set is the one that we used for our results. From the parameter marginalisation it was possible to produce a posterior prob-ability density distribution for the coupling level as shown in Figure 8. The cou-pling level predicted from the distribution is bounded between 0 and 1:17000with 95% confidence, with a mean coupling level of 1:27600. The probabil-ity density distributions for the scaling and standard deviation parameters areshown in Figure 9. The scaling posterior distribution indicates a mean valueof 29.3 × − with standard deviation 0.94 × − . The posterior distributionfor the beam smearing parameter indicates a range of possible values between0 and 1.3 × − m.The measured cavity length signals as well as the 95% upper limit and meanWGM coupling level predicted by the analysis are shown in Figure 10. The phasediscrepancy between the model and the measurements, as witnessed in thisfigure most profoundly for the spot positions around − × − m, is thoughtto be an artefact from the modelling of the beam smearing effect. The residual “Yet Another Matlab MCMC code” by Matthew Pitkin. Available as of time of writingat https://github.com/mattpitkin/yamm . nd order Littrow grating measured in [32], where the coupling factor was of order1:100. P r o b a b ili t y D e n s i t y [ a r b . un i t s ] Figure 8: Posterior probability density distribution of WGM coupling levels (inunits of meters longitudinal per metre transverse) yielded by statistical analysisof the data. The red shaded region shows the coupling levels falling within themost probable 95% of the distribution.16 P r o b a b ili t y D e n s i t y [ a r b . un i t s ] P r o b a b ili t y D e n s i t y [ a r b . un i t s ] Figure 9: Posterior probability density distribution of other parameters used inthe analysis: scaling applied to the model’s predicted longitudinal signal (leftplot) and the standard deviation assumed for the Gaussian distribution used tomodel beam smearing (right plot). Both distributions lie well within their priorranges (see Table 4). 17 -4 -3 -2 -1 C a v i t y l e n g t h c h a n g e [ m / r a d ] Model, 1:27600Model, 1:17000Model, no couplingMeasurements P h a s e [ ° ] Figure 10: Measurements and simulations of the cavity length signal for spotpositions with respect to the ETM’s centre of yaw. The calibrated cavity lengthchange per radian (vertical axis) from the measurements is shown (blue stars)alongside the model’s simulated cavity length changes per radian for the mean(red), 95% upper limit (green) and zero (black) WGM coupling levels. Thesimulated plots use a scaling factor of 29.3 × − and a beam smearing standarddeviation of 0.8 × − m.Error bars are shown on the measured spot positions corresponding to theiruncertainty. The errors in cavity length change are obtained from the noise floorsurrounding each measurement. The noise floors were approximately constantfor all measurements, with mean value 8 × − m rad − . Phase error bars arevisible for the central values. The errors on each phase measurement, fromleft to right, are: +/-0.0188, +/-0.0254, +/-0.0283, +/-0.1387, +/-0.1721, +/-0.2178, +/-3.2726, +/-3.2303, +/-2.0603, +/-0.0385, +/-0.0342 and +/-0.0336degrees. The authors would like to thank members of the LIGO Scientific Collabora-tion for fruitful discussions. The Glasgow authors are grateful for the supportfrom the Science and Technologies Facility Council (STFC) under grant numberST/L000946/1. The Jena authors are grateful for the support from the DeutscheForschungsgemeinschaft under project Sonderforschungsbereich Transregio 7.
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