The Advanced Virgo Photon Calibrators
D. Estevez, P. Lagabbe, A. Masserot, L. Rolland, M. Seglar-Arroyo, D. Verkindt
TThe Advanced Virgo Photon Calibrators
D. Estevez , , P. Lagabbe , A. Masserot , L. Rolland , M.Seglar-Arroyo , D. Verkindt Institut Pluridisciplinaire Hubert CURIEN, 23 rue du loess - BP28 67037 Strasbourgcedex 2, France Laboratoire dAnnecy de Physique des Particules (LAPP), Univ. Grenoble Alpes,Universit Savoie Mont Blanc, CNRS/IN2P3, F-74941 Annecy, FranceE-mail: [email protected]
Abstract.
As the sensitivities of LIGO, Virgo and KAGRA detectors improve, calibrationof the interferometers output is becoming more and more important and may impact scientificresults. For the observing run O3, Virgo used for the first time photon calibrators (PCal)to calibrate the interferometer, using radiation pressure of a modulated auxiliary laser beamimpinging on the Advanced Virgo end mirrors. Those optical devices, also used in LIGO,are now the calibration reference for the global gravitational wave detectors network. Theintercalibration of LIGO and Virgo PCals, based on the same absolute reference called theGold Standard, has allowed to remove a systematic bias of 3 .
92% that would have beenpresent in Virgo calibration using the PCal. The uncertainty budget on the PCal-induceddisplacement of the end mirrors (NE and WE) of Advanced Virgo has been estimated to be1 .
35% for O3a and 1 .
39% on NE PCal (resp. 1 .
73% on WE PCal) for O3b. This uncertaintyis the limiting one for the global calibration of Advanced Virgo. It is expected to be reducedbelow ∼
1% for the next observing runs. a r X i v : . [ a s t r o - ph . I M ] S e p
1. Introduction
A gravitational wave interferometric detector like Advanced Virgo needs a stable workingpoint, close to a dark fringe, to detect light variations induced by a gravitational wave [1].To keep the interferometer on this working point, longitudinal control loops are used, thatinvolve actuators on the mirrors of the interferometer. The reconstruction of the gravitationalwave strain h ( t ) from the dark fringe light power variations requires a precise and accuratemeasurement of those actuators response which is the aim of the calibration [2].The Advanced Virgo detector participated in August 2017 to the observation run O2and allowed the first triple coincident detection of a binary black holes merger [3] and the firstdetection of a binary neutron stars merger with electromagnetic counterpart [4][5]. Duringthis period, the calibration was using the input laser wavelength as a reference to measurethe mirrors electromagnetic actuators response used in the reconstruction of the gravitationalwave signal [2][6]. For the O3 observation run, from April 1st 2019 to March 27th 2020, theimproved sensitivities of the Advanced LIGO and Advanced Virgo detectors required toreduce the calibration uncertainties on h ( t ). This motivated the use in Virgo of PhotonCalibrators (PCal) as reference to measure the mirrors actuators response [7]. Moreover,PCals are also used on the Advanced LIGO and KAGRA detectors, hence an intercalibrationof the global gravitational wave detectors network based on the same reference could beperformed [8][7].After an introduction describing the principle of the PCal, this paper gives in section 2a detailed description of the PCals that have been implemented on the Advanced Virgointerferometer and describes in section 3 the first work done on PCal intercalibrationbetween Virgo and LIGO. Then, section 4 presents the uncertainty budget on the end mirrordisplacement induced by an Advanced Virgo PCal, estimated over the full observing runO3. Eventually, the conclusion in section 5 contains also a discussion about the challenge ofgravitational wave detectors calibration for the next observing runs. The aim of the PCal is to use the radiation pressure of a modulated laser beam, whoseoutput power is well known and controlled, to induce on an end mirror of the interferometera displacement which translates into a modification of the dark fringe signal at the outputport of the interferometer and a measured modification of the reconstructed equivalentgravitational wave strain signal h ( t ).The force induced by radiation pressure on the end mirror with a PCal is expressed as: F pcal ( f ) = 2 cos( θ )c P end ( f ) (1)with θ the angle of incidence of the PCal laser beam impinging on the end mirror, c thespeed of light and P end ( f ) the laser power reflected by the end mirror at the modulatedfrequency f . This force generates a displacement of the end mirror which is governed by themechanical response of the suspended optic. This mechanical response is well approximated(within ± . . m (2 πf ) ] − , with m themass of the optic. The induced displacement is thus: x freepcal ( f ) = − m (2 πf ) θ )c P end ( f ) (2)As the PCal system is acting on an end mirror of the interferometer, the laser powernoise introduces some unwanted displacement at any frequency, thus an additional noisein the sensitivity of the interferometer. Therefore, a digital system to mitigate the laserpower noise of the PCal and to stabilize the modulated output power has been implemented,as described in section 2.2, so that the remaining broadband noise does not significantlycontribute to the Advanced Virgo sensitivity. In practice, Eq. 2 works well for the Advanced Virgo PCal between 10 Hz and 400 Hz. Indeed,contrary to the Advanced LIGO PCal two beams configuration [9], the Advanced VirgoPCal uses a single laser beam impinging the center of the end mirror of the interferometer.Resonant axisymmetric elastic modes of the optic are thus excited and their contributionaffects the mechanical transfer function of the PCal above 400 Hz as it has already beendemonstrated in [10]. In Advanced Virgo, the drumhead mode of the end mirror measuredaround 7813 Hz is expected to have a significant contribution to the mechanical response ofthe PCal in the frequency range 400 Hz to 2 kHz. The coupling of the drumhead mode withthe displacement of the end mirror can be modeled as a second order low-pass filter given by: H d ( f ) = G d jQ d ff d − (cid:16) ff d (cid:17) (3)with G d the gain of the drumhead mode coupling, f d the resonant frequency of the modeand Q d the quality factor. The effective displacement of the end mirror sensed by theinterferometer is thus driven by the following equation: x pcal ( f ) = (cid:104) − m (2 πf ) + H d ( f ) (cid:105) θ )c P end ( f ) (4)For convenience we also define the PCal induced strain as: h pcal ( f ) = x pcal ( f ) L (5) Free mass response Drumhead mode coupling responsePcal response
Figure 1: Expected amplitude of the mechanical response of the Advanced Virgo Pcal instrain per watt unit. This PCal response is the complex sum of the free mass response andof the contribution of the drumhead mode excitation of the end mirror at 7813 Hz.where L = 3000 m is the nominal length of one arm of the Advanced Virgo interferometer.Figure 1 shows the expected transfer function from the reflected PCal laser power P to theinduced end mirror strain h pcal . As the contributions of the free mass response and of thedrumhead mode coupling response are in phase opposition between 10 Hz and 7813 Hz, anotch is present at 2050 Hz in the PCal response where the amplitude of the two contributionsare equal. This means that the interferometer will not be able to sense any displacement ofthe end mirror induced by the PCal at this frequency. For frequencies above the notch, thePCal response is enhanced by the drumhead mode coupling instead of falling as ∝ f − as itis the case for a free mass.
2. Experimental setup
Two photon calibrators have been installed at the West End (WE) and the North End (NE)stations of the Advanced Virgo interferometer. In addition to being used as reference for thedetector’s calibration, they allow the verification of the reconstruction of the gravitationalwave signal as discussed in section 5.As shown in Figure 2, each PCal setup is composed of two optical benches. The injectionbench is used to send the laser beam, stabilized in power by a fast digital control loop, tothe inner cavity surface of the end mirror. The reflection bench is used to measure the powerreflected by the end mirror with a Si photodetector sensitive over 1 cm . The laser beam hitsthe center of the end mirror with an angle of incidence θ of 18 . ◦ . ReflectionBench(in air)InjectionBench(in air)
End mirrorMain interferometer beam PCal beam(in vacuum) θ Mass m Figure 2: Schematic of an Advanced Virgo photon calibrator viewed from the top.
As shown in the optical sketch of the injection and reflection benches of Figure 3, theAdvanced Virgo PCal laser beam is generated with a diode laser source at 1047 nm. This isthe same wavelength as the one used for the Advanced LIGO PCals, which makes easier thelaser power intercalibration of both detectors. The PCal laser wavelength is close enough tothe main interferometer beam wavelength at 1064 nm which ensures a high reflectivity of theend mirror of the interferometer but it is also different enough so that PCal scattered lightdoes not introduce additional noise in the interferometer.The range of deliverable power goes from 0 to 3 W hence the PCal operates at 2 Wto be able to modulate the power up to ± ≥ . ∼ in-loop sensor used for the Fast DigitalControl Loop described in section 2.2 and its calibration is detailed in section 4.1. It is alsopossible to monitor the optical leverage with a position sensitive detector named PSD1 butit was not used during O3.The calibration of the photodetectors can be affected by environmental variations oftemperature and humidity. Therefore, a thermal sensor and a hygrometer ‡ have beenimplemented on the injection bench close to PD1 for monitoring.The reflection bench houses detectors similar to the ones of the injection bench. ThePCal laser beam reflected by the end mirror of the interferometer reaches the reflection benchthrough a viewport similar to the one of the injection bench. Only a small fraction of thisbeam ( ∼ (1PPS) signal onto PD2. Indeed,the photodiode signal is digitally processed and sampled at 20 kHz which induces a delayin the readout. Measuring the delay of the 1PPS signal in the 20 kHz channel allows tocalibrate the timing of the PCal, thus the timing of the PCal-induced end mirror motion ofthe interferometer. The Advanced Virgo PCal operates with an input power of 2 W. In this state, inherentfluctuations of power in the laser beam occur at every frequency, inducing a broadbanddisplacement of the end mirror. This laser power noise, converted into strain noise, limits ‡ The thermal sensor has been implemented at the beginning of O3a (from April 1st 2019 to September 30th2019) and the hygrometer at the beginning of O3b (from November 1st 2019 to March 27th 2020.)
LaserBeam dump Polarizing beamsplitter cubeThermal sensor and hygrometerMirrorViewport PSD1PD1LensPD2PSD2
Reflection bench Injection benchMain interferometer beam T o w a r d e n d m i rr o r LED
Figure 3: Detailed schematic of an Avanced Virgo photon calibrator viewed from the top.The size of the benches is 40 cm ×
40 cm.the sensitivity of the Advanced Virgo interferometer. Therefore, it has been mitigated tolimit its contribution below 10% of the O3 sensitivity.A fast digital control loop has been implemented to satisfy the above requirement bystabilizing the laser output power at 2 W. This loop is handled by a real-time process runningat 200 kHz and using an in-loop output power signal witnessed by the photodiode locatedon the injection bench. The open-loop and closed-loop transfer functions that characterizethe fast control loop have been measured (see Figure 4). The unity gain frequency is close to4 . ◦ which ensures a robust control of the system. At 1 kHzthe discrepancy between the requested signal and the output signal is − . − τ = 81 µ s ( phase = − ◦ ).The control loop has been running permanently during O3 to stabilize the output powerof the laser and to mitigate the laser power noise. Figure 5 shows the PCal laser powernoise with and without the control loop. Thanks to the loop, the laser power noise duringthe O3 run was more than one order of magnitude below the requirement. It is also worthmentioning that there are three spectral lines remaining above the requirements. One ofthem is the 50 Hz signal coming from the distribution of the mains. The two other linesare permanent sinewave excitations at 36 . . A m p li t ude [ d B ] P ha s e [ deg ] Frequency [Hz]Frequency [Hz]
Open Loop TFClosed Loop TF
Figure 4: Open-loop (blue) and closed-loop (red) transfer functions of the PCal Fast DigitalControl Loop. The measured unity gain frequency is around 4 . ◦ . gr1 Entries 100000Mean 4417RMS 2621
Frequency [Hz]10 La s e r P o w e r N o i s e [ W / s q r t ( H z ) ] -7 -6 -5 -4 -3 -2 gr1 Entries 100000Mean 4417RMS 2621 gr1
Entries 100000Mean 4417RMS 2621
Figure 5: Laser power noise of the Advanced Virgo PCal without control loop (blue) andwith control loop (black). The requirement for the laser power noise of the PCal contributionat 10% of the sensitivity for the observing run O3 is shown in red.
The estimation of the laser power reflected by the end mirror is done using the photodiodeson the injection and reflection benches. To do so, those photodetectors have to be carefullycalibrated so that the voltage delivered by the sensor receiving the beam can be translatedinto effective laser power unit reflected by the end mirror. The method for this calibrationprocedure is to compare laser power measurements with an integrating sphere, so-called VirgoIntegrating Sphere (VIS), on the injection and reflection benches recording the laser beamgoing into the vacuum tower and the laser beam going out of it. In the same time, thephotodiode on the injection bench (PD1) collects a fraction of the light of the laser beamand delivers a certain voltage which is recorded and which can then be converted into powerunit using the averaged results of VIS measurements from both benches. Once PD1 has beencalibrated, VIS is no longer used to calibrate PD2, and the calibration of PD2 is performedcomparing the calibrated laser power measured by PD1 against the voltage delivered by PD2.Before performing the photodiodes calibration, one needs to be sure that VIS powercalibration is absolute . In section 3, the absolute calibration of VIS is described and thissphere has been used during O3 as the calibration reference for the PCal sensors.
3. Intercalibration with LIGO
One of the main challenge to estimate the displacement of an end mirror of Advanced Virgoinduced by a PCal is to determine the PCal laser power reflected by the end mirror. Theaccuracy on this laser power is the limiting factor of the calibration of the interferometerand impacts the precision of the reconstructed gravitational wave strain provided to dataanalysis. A pick-off of the reflected laser beam is sensed by photodiode PD2 on the reflectionbench and has to be calibrated in an absolute manner so that the voltage delivered by thesensor can be precisely converted into absolute power reflected by the end mirror. Thecalibration of the photodiode is done using the Virgo Integrating Sphere (VIS) which isa Newport 3 . ± .
4% in the range0.2 W to 3 W. The method to derive the conversion factor of the photodiode from voltageto power is to simultaneously record the PCal laser power with the integrating sphere andthe output voltage of PD1, as described in section 2.3. PD2 is then calibrated against PD1after having removed the integrating sphere of the bench and using the laser beam hittingsimultaneously both photodiodes. This procedure requires an absolute calibration of theVirgo integrating sphere by carefully chosing an absolute calibration reference which is theLIGO Gold Standard (GS) calibrated by the National Institute of Standards and Technology(NIST) in Boulder, CO. [11].Since LIGO and Virgo are performing a coincident analysis of the calibrated gravitational0wave data stream provided by each interferometer, one has to be sure that the relativecalibration between each detector does not introduce any bias in the analysis or that atleast the putative bias is as small as possible. During the observing run O2, the FreeSwinging Michelson technique was used as a reference for Virgo calibration and could notbe directly compared to LIGO calibration based on the PCal [12]. Indeed, the absolute reference for LIGO was the Gold Standard and the one for Virgo was the wavelength of themain interferometer laser beam. The decision to use the PCal on Virgo for the observing runO3 was then motivated by the different upgrades performed on the setup which allowed to beconfident on its calibration, stability and precision. The use of the PCal was also motivatedby the possibility to intercalibrate the PCal laser power between LIGO and Virgo with theGold Standard. Figure 6 shows the calibration chain from GS to the PCal power sensorslocated on the injection and reflection benches of the Advanced Virgo interferometer and alsothe optical setup to calibrate VIS against GS.
Gold Standard(NIST) Working Standardfor Virgo(LHO/LAPP labs)Virgo IntegratingSphere(LHO/LAPP labs)Injection and Reflection benchesPCal power sensors(Virgo Cascina) (a)
PCal lasermodule
GSVIS
Pneumatic sliders
BSM1M2 M3 ‘r’ beam‘t’ beam (b)
Figure 6: (a) Diagram of the calibration chain made for the laser power calibration of theAdvanced Virgo PCal. The LIGO Gold Standard calibrated by NIST stands at the topof the chain and serves as absolute reference for the Advanced Virgo PCal power sensorsat the bottom of the chain. The location given in brackets indicate where the calibrationmeasurements were performed. (b) Schematic of the optical setup used to calibrate the VirgoIntegrating Sphere against the Gold Standard. A PCal laser module is used to generate alaser beam which is then split into two beams with a beamsplitter (BS). The Gold Standardand the Virgo Integrating Sphere are mounted on pneumatic sliders to swap between thereflected (r) and transmitted (t) beams to determine the ratio of the responsivities. Thissetup is also used to calibrate the Working Standards of Advanced LIGO and KAGRA.1
The calibration of VIS against GS consists in measuring:Γ
V IS/GS = ρ V IS ρ GS (6)which is the ratio of the integrating spheres responsivities ρ . The calibration factor Γ V IS/GS has been computed as follows:Γ
V IS/GS = (cid:115) ( P V IS,r − P BGV IS,r ) · ( P V IS,t − P BGV IS,t )( P GS,r − P BGGS,r ) · ( P GS,t − P BGGS,t ) (7)with r and t standing for the reflected and transmitted beam respectively which denotes theposition of the spheres on the pneumatic sliders from Figure 6(b). P stands for the measuredpowers with incoming laser beam and BG indicates the background measurements with thelaser turned off that are subtracted to the measured laser power values. Doing ratios of powermeasured by GS and VIS eliminates simultaneous laser power variations and swapping theirposition eliminates the effect of beamsplitter imperfections. As a result this procedure givesaccess to the ratio of the integrating spheres responsivities ρ .The calibration factor has been computed for five series of measurements performed atLHO in February 2019 on five different days and the results are shown in Figure 7. Theaverage of those measurements is Γ V IS/GS = 0 . ± .
1% taking into account thedispersion of the points as a systematic uncertainty. The uncertainty on the linearity of thereadout of VIS ( ± . absolute calibration factor measured byNIST for GS ( ± . V IS/GS = 0 . ± . V IS/GS . Since VIS is used on Virgo to calibrate the PCal it cannot be compared against GS (whichstays at LHO) as often as needed to check the stability of the calibration factor. A WorkingStandard for Virgo (WSV) similar to the Working Standards used on LIGO and KAGRAhas thus been mounted at LHO with the aim of staying at LAPP § , where the Virgo PCalis developed, so that VIS could be compared against WSV during O3. This integratingsphere WSV was calibrated against GS with the same setup used to calibrate VIS. Figure 8shows the calibration factor corresponding to the ratio of the responsivities of WSV and GScomputed as in equation 7 for six series of measurements performed at LHO in February § Laboratoire d’Annecy de Physique des Particules P ∼ . W SV/GS = 0 . ± . P ∼ . The ratio of the responsivities of VIS and WSV has also been measured at LHO in the sameexperimental conditions as for the previous ratios described in the above sections. The valueof this ratio is the reference from which the stability of VIS calibration is estimated andhas to be monitored at LAPP during O3. Therefore, a similar optical setup to calibrate theintegrating spheres as the one at LHO has been mounted at LAPP during O3 to check thestability of VIS calibration. Since the acquisition tools are different from LHO to LAPP weused a voltage calibrator to calibrate our voltage readout at LAPP at the level of 0 . V IS/W SV performed at LHO in February 2019and the five measurements performed at LAPP in June and October 2019. The dispersionof the points around the mean value is ± .
5% which is thus the value used during O3 tocharacterize the stability in time of the intercalibration between LIGO and Virgo.Investigations to understand the systematic uncertainties related to these measurementsare needed for the next observation run O4 in order to find solutions to improve the stability intime of the intercalibration. More measurements of Virgo spheres against the Gold Standardwill also have to be performed to strengthen the confidence in the calibration factors.Figure 9: Calibration factors between the Working Standard for Virgo and the VirgoIntegrating Sphere. The red points have been measured by averaging 30 sets of 100 smeasurements at P ∼ . P ∼
4. Uncertainties
The total uncertainty on the end mirror displacement induced by the PCal arises from thedetermination of all the parameters from equation 4. The main contribution to this totaluncertainty comes from the estimation of the laser power reflected by the end mirror. Indeed,many factors have to be taken into account from the absolute calibration of VIS with GSdown to the Advanced Virgo PCal sensors calibration.
The power measurements on the injection and reflection benches with VIS to calibrate thephotodiodes are done outside the vacuum tower containing the end mirror. The laser beam isthus affected by the optical losses of the viewports between the tower and the PCal benches.Hence, to precisely estimate the power reflected by the ETM, the viewport losses have to becharacterized.The viewports are coated on both in-air and in-vacuum surfaces with 1064 nm broadbandanti-reflective coatings whose power reflectivity had been estimated to be around R = 0 . .
99% and is thus not affecting the total optical efficiency of the PCal. Therefore, the powerreflected by the end mirror P end can be expressed as: P end = P inj (1 − R ) = P ref (1 − R ) (8)where P inj and P ref are the laser power respectively measured on the injection and reflectionbenches. P end can thus be approximated, at the first order, as the average of P inj and P ref . This is what has been used for the photodiodes calibration. We expect the opticalefficiency η = P ref /P inj to be around 0 .
998 considering the viewport losses meaning that theuncertainty on P end should be within 0 . . . .
3% higher. The measurements were thus performed with VIS located where the beam sizeis the largest on the injection bench, and the laser power was then 0 .
3% higher than thepower measured on the reflection bench which is closer to the value we would expect fromthe optical efficiency.Since we could not estimate properly the optical efficiency of the PCal due to the effectmentioned above, we decided to keep a conservative uncertainty of 1% for this measurementresulting from 0 .
8% due to power variations depending on VIS position plus 0 .
2% consideringthe expected optical efficiency. This is the dominant uncertainty in the final error budgetof the PCal and it will have to be tackled and characterized more accurately in the future.A possible study would be to measure the laser power with VIS at different positions on anoptical bench and with different laser beam sizes.Table 1 summarizes the sources of systematic uncertainties on the estimation of thePCal laser power reflected by the end mirror of the interferometer, from the top (the GoldStandard) to the bottom (the PCal sensors) of the laser power calibration chain. Then, thetotal relative uncertainty on P end is computed as: σ P P end = (cid:104) (cid:88) i (cid:16) σ x i x i (cid:17) (cid:105) / (9)with x i the different parameters estimated as sources of systematic uncertainties in the laserpower calibration chain. Parameter 1 σ UncertaintyGS responsivity (2018) 0 . . . . . . The geometrical parameters are also contributing to the overall uncertainty on the end mirrordisplacement. Above the resonant frequency of the end mirror suspension, the displacementof the optics induced by a PCal is inversely proportional to the mass of the optics. In Ad-vanced Virgo, the mass has been measured to 42 . ± .
02 kg which gives a 1 σ uncertainty6of ± .
05% on the PCal-induced end mirror displacement.The angle of incidence of the PCal laser beam hitting the end mirror has been evaluated,with optomechanical constraints from the drawings, to be θ = 18 . ◦ . This angle is limitedby the diameter of the viewports. This diameter is 63 mm and the beam has been centeredon the viewports better than ±
10 mm. The 1 σ uncertainty on the angle of incidence is thustreated as a Type-B uncertainty [13] contributing in the cosine as ± . x pcal ( f ) = − m (2 πf ) θ )c H ( f ) P ( f ) (cid:16) (cid:126)a · (cid:126)b mI (cid:17) (10)where I is the rotational moment of inertia of the end mirror of mass m and (cid:126)a (resp. (cid:126)b ) is thevector from the center of the mirror to the position of the main interferometer laser beamspot (resp. PCal laser beam spot) on the optic. In Advanced Virgo, the centering of themain interferometer beam is controlled to be better than ± . ±
20 mm due to optomechanical constraints. Considering the worstcase scenario where the scalar product between the two vectors is extremum (both beamspots are shifted in the same direction), the miscentering of the beams leads to a relativeerror of ± . σ uncertaintyMass of the end mirror 0 . . . . As mentioned in previous sections, the PCal laser beam hits the center of the end mirror ofthe interferometer and thus excites axisymmetric internal modes of the optic. The model ofthe PCal-induced motion of the end mirror is given by Eq. 4 where H ( f ) is the last item7 center ⃗ a ⃗ b Suspended mirror
ITF beamPCal beam (a)
60 40 20 0 20 40 60Photon calibrator beam offset b (mm)0.01000.00750.00500.00250.00000.00250.00500.00750.0100 E rr o r f a c t o r f r o m t il t ( % ) Relative error on mirror displacement due to beam misalignment a = 0 mma = 0.5 mma = -0.5 mma = 2 mm (b)
Figure 10: (a) Schematic of a suspended end mirror of the interferometer with the maininterferometer (ITF) laser beam and the PCal laser beam spots shifted from the center ofthe optics by (cid:126)a and (cid:126)b respectively. The miscentering of the beams is exagerated in this figurefor clarity purpose. (b) Relative error on the end mirror displacement due the tilt of theoptics induced by a miscentering of the main interferometer beam and the PCal beam. Thevectors positioning the beams are assumed to be collinear in order to maximize the error.of the equation that we have to characterize. The value of the drumhead mode f d has beenmeasured to be 7812 . ± . . ± . .
88 Hz / ◦ C and by consideringthe reference temperature during the measurements of f d . The maximum deviation oftemperature from the reference temperature during the measurements is 1 . ◦ C for WE and0 . ◦ C for NE. These temperature variations lead to a relative uncertainty on f d of ± . ± . Q d is of the order of 10 whichis high enough so that an error of a few percent on this value has a negligible contributionon the global uncertainty of H ( f ).The last parameter for which an uncertainty has to be estimated is the gain of the drumhead mode G d . The measurement of this gain has been done by measuring the frequencyof the notch in the mechanical response of the PCal around 2 kHz as shown in the simulatedresponse in Figure 1. Indeed, the notch is the result of the free-mass response having thesame amplitude as the internal deformation of the mirror but in phase opposition. The idea8is thus to compare a PCal-induced strain h pcal on the end mirror, taking into account only thesimple pendulum model H p , with the reconstructed strain of Advanced Virgo interferometer h rec . Taking the transfer function from the PCal strain to the reconstructed strain will revealthe discrepancy between both strains which arise from the unmodeled drumhead mode. It isimportant to notice that this measurement depends on the reconstruction of the gravitationalwave strain and may introduce unwanted bias in the measurement. We thus assume that ifthere is a bias in h rec around the notch frequency band it is a constant bias on the amplitudeand it is not frequency dependent so that the shape of the measurements is unchanged. Thequantity that we want to measure and fit can be expressed as: h rec h pcal ∝ H p + H d H p ∝ G − f λ (cid:18) − (cid:16) ff d (cid:17) (cid:19) (11)(12)where G and λ are the two parameters of the fit. The parameter λ is linked to the notchfrequency f n by the following relation: f n = λf d (cid:112) λ + f d (13)Since the gain of the drumhead mode coupling response at the frequency f n is not equalto the static gain ( f = 0) of the coupling, we have introduced λ which is the equivalentfrequency where the amplitude of the free-mass response is equal to the static gain of the drumhead mode coupling. Fitting for λ instead of f n gives us directly the static gain G d inmeter per newton expressed as: G d = 14 π mλ (14)Figure 11 shows the amplitude of the transfer function for both WE and NE PCals. Thegain of the drumhead mode coupling response for both PCals have been renormalized in unitof strain per watt for convenience and have been estimated to: G W Ed = (2 . ± . × − h/W G NEd = (2 . ± . × − h/WThe parameters of the drumhead mode coupling for both end PCals are then known and theresulting uncertainties are gathered in Table 3. One can notice that those uncertainties adda frequency dependent contribution to the global uncertainty budget of the PCal-inducedend mirror motion.9 Frequency [Hz] G a i n [ a . u .] / ndf c G 0.001007 – l – c G 0.001007 – l – Notch measurement on WE mirror with the PCal (a) WE
Frequency [Hz] G a i n [ a . u .] / ndf c G 0.001068 – l – c G 0.001068 – l – Notch measurement on NE mirror with the PCal (b) NE
Figure 11: Measured amplitude of h rec /h pcal for WE PCal (a) (NE PCal (b)) and theassociated fit. PCal ∆ G d ∆ f d WE ± . ± . ± . ± . G d and f d for WE and NE PCal mechanical responses. The calibration of the PCal photodiodes has been monitored during O3 to look for anyother sources of systematic uncertainties and to check the calibration stability in time.The responsivity of the Advanced Virgo PCal photodiodes depends on the temperatureas 0 . / ◦ C at 1047 nm. A monitoring of the surrounding temperature during O3 hasthus been done to evaluate the impact on the overall PCal uncertainty budget. Sincethe temperature variations do not follow a gaussian distribution we treated the changein photodiode responsivity as a Type-B uncertainty assuming a rectangular distribution oftemperatures over the whole range of variations. The highest range of temperature variationswas found to be 0 . ◦ C on NE PCal reflection bench which resulted in a Type-B uncertaintyof ± .
1% on the photodiode calibration, to be added to the PCal uncertainty budget.Some variations of photodiodes power calibration larger than the expected ones due totemperature variations were seen between PD1 and PD2 during O3a as shown in Figure 12.The monitoring and the analysis of these variations were done using calibration lines ofthe PCal. Those lines are sinewave excitations sent to the end mirrors of the interferometerby modulating in amplitude the laser beam of the PCal. A long term study was done overO3a comparing the reconstructed signal h rec from the output of the interferometer against0 (a) WE W / W (b) NE Figure 12: Ratio of PD1 and PD2 signals over O3a at the frequency of the calibration lines.(a) The calibration line frequency is 60 . . h pcal reconstructed from the PCal photodiodes. Figure 13 shows the ratio of h rec over h pcal for both photodiodes on both PCals during O3a. The photodiodes PD2 on WEand PD1 on NE are the two photodiodes that contribute the most to the variations seen onFigure 12. Only the photodiodes PD1 on WE and PD2 on NE were thus used to estimate thePCal-induced end mirrors motion during O3. The uncertainty on their calibration stabilityhas been assessed using the width of the distributions of h rec over h pcal . Figure 14 showsthese distributions for O3a and O3b. Those distributions account for calibration variationsof h pcal but also the ones of h rec , thus only an upper limit on the uncertainty on the h pcal stability can be drawn from this analysis. Therefore the 1 σ uncertainty on the stability ofthe photodiodes power calibration has been conservatively estimated to ± .
5% over the O3aperiod.In between O3a and O3b (cid:107) , the NE and WE driver lasers had to be repaired after anelectrical failure. Only the WE driver laser was mounted back in time for the start of O3band the NE driver laser has been reinstalled later during O3b in January 2020. A few daysbefore O3b, the WE PCal set-up had to be realigned and WE PD1 recalibrated. The newmeasured calibration factor for WE PD1 differed by +1 .
3% from the one of O3a. Thisdifference exceeds the uncertainty of 0 .
8% stated in Section 4.1 due to VIS positioning onthe optical bench and therefore was significant enough to be corrected for O3b. One can seein Figures 14(a) and 14(c) that the mean values of the distributions differ by ∼ .
7% andwith similar standard deviations of 0 . .
5% for O3a to (cid:107)
One month break in the observation run of LIGO and Virgo from October 1st to November 1st 2019. .
2% for O3b as explained further in this section.NE driver laser was mounted back on the 21st of January 2020 but not recalibrated tobe able to compare the new set-up to the previous one and also with WE PCal. Figures14(b) and 14(d) show that the mean values between both distributions differ by 0 .
5% withsimilar standard deviations of 0 . . other sources that affect the PCal laser power stability. (a) WE PD1 (left) and WE PD2 (right). (b) NE PD1 (left) and NE PD2 (right). Figure 13: Ratio of h rec and h pcal for both PD1 and PD2 signals on both PCals at thefrequency of the calibration lines during O3a. (a) Calibration line frequency is 60 . . other sources of variations which affect the stability intime of the PCal calibration have shown that the relative humidity variations around the2 (a) WE PD1 O3a (b) NE PD2 O3a(c) WE PD1 O3b (d) NE PD2 O3b Figure 14: Distribution of h rec /h pcal using (a) WE PD1 and (b) NE PD2 during O3a and(c) WE PD2 and (d) NE PD2 during O3b. The standard deviation of the distributionsis ∼ .
5% and gives an upper limit on the stability of the PCal calibration over O3a andO3b for a given calibration of the photodiodes.PCal benches are correlated with the photodiodes calibration variations.
Relative humidity variations affecting the PCal calibration
We have observed that the variations of humidity inside the NE PCal bench was corre-lated with the NE PD1 signal during O3a and O3b as shown in Figure 15. The correlationvaries in time and the behaviors during O3a and O3b are different. During O3a, at least twobands of correlation can be seen whereas during O3b a phenomenon of hysteresis has been3Parameter 1 σ uncertainty O3a 1 σ uncertainty O3bNE WE NE WEResponsivity (temperature) ± . ± . ± . ± . ± . ± . ± . ± . .
51% 0 . ± . ± . ∼ . .
6% uncertainty estimated for the stability in time of its calibration over O3a and O3b.Similar investigations have been performed on WE PCal photodiodes and are shown inFigure 16. During O3a, WE PD1 signal variations due to humidity changes are consistentwith the 0 .
5% uncertainty given for its calibration stability in time. It is also noticeable thatthe main part of the 1 .
5% variations of WE PD2 signal during O3a is not correlated withhumidity changes and that humidity variations may count only for 0 .
5% of the WE PD2photodiode calibration variations. Regarding O3b, the variations of WE PD2 signal due tohumidity changes were also within ∼ . . h rec /h pcal differed by 0 .
5% with the one of O3a at the same relativehumidity (see Figures 16(a) and 16(c)). This indicates that the calibration measurements ofWE PD1 performed before O3a and a few days before O3b are coherent and compatible withthe uncertainty of 0 .
8% on VIS positioning on the optical bench. However, during O3b, WEPD1 calibration started to drift by ∼ .
1% due to humidity variations. Adding quadraticallythe uncertainty of 0 .
5% around the mean value, the uncertainty on WE PCal calibrationstability for O3b has been increased to 1 . Below the notch frequency, the total uncertainty on the NE (resp. WE) photon calibratorshas been estimated to be 1 .
35% (resp. 1 . .
39% (resp. 1 . (a) NE PD1 O3a (b) NE PD2 O3a(c) NE PD1 O3b (d) NE PD2 O3b Figure 15: Distribution of h rec /h pcal for NE PCal photodiodes as a function of relativehumidity (R.H.) surrounding NE PCal during O3a and O3b.Parameter 1 σ uncertainty O3a 1 σ uncertainty O3bNE WE NE WEReflected laser power ( P ) 1 .
24% 1 .
24% 1 .
24% 1 . .
13% 0 .
13% 0 .
13% 0 . .
51% 0 .
51% 0 .
61% 1 . .
35% 1 .
35% 1 .
39% 1 . (a) WE PD1 O3a (b) WE PD2 O3a(c) WE PD1 O3b (d) WE PD2 O3b Figure 16: Distribution of h rec /h pcal for WE PCal photodiodes as a function of the relativehumidity (R.H.) surrounding WE PCal during O3a and O3b.be 110 ± µ s. This delay has been corrected to get the absolute timing of the PCal-inducedend mirror motion during O3.
5. Conclusion
The photon calibrators developed since several years in Virgo and used in a preliminaryversion during O2, have been improved and used for the first time as a calibration referenceduring the observation run O3. This allowed to put the relative calibration of the gravitationalwave detectors network on the same absolute calibration reference: the Gold Standard.The first work of intercalibration between Virgo and LIGO PCals allowed to correct fora discrepancy of 3 .
92% on the measured laser power between the detectors. A WorkingStandard for Virgo, similar to the Working Standards used in LIGO and KAGRA, has alsobeen mounted to check the stability in time of the Virgo Integrating Sphere calibration.6Figure 17: Frequency dependent total uncertainty on the NE and WE end mirrorsdisplacement induced by the photon calibrators for O3a.On the Advanced Virgo PCal, the laser power has been digitally controlled in orderto keep its broadband noise contribution more than 10 times below the sensitivity of theAdvanced Virgo interferometer. In addition, the systematic uncertainty on the PCal-inducedend mirror motion of the interferometer has been estimated to be 1 .
35% for both PCal duringO3a and 1 .
39% (resp. 1 . . injection bench with a waist located on the interferometer’s end mirror so thatthe beam on the injection and reflection benches has approximately the same size. Moreover,the Si photodiodes used to estimate the laser power reflected by the end mirror should bereplaced by InGaAs photodiodes whose responsivities would have a smaller dependency ontemperature variations on the PCal benches. During O3a, the stability of the photodiodescalibration has been estimated to 0 .
5% on WE and NE PCals but bigger variations, correlatedto humidity variations, have been seen on one of the NE photodiode on the injection bench.During O3b, NE PD2 calibration stability was updated to 0 .
6% with variations due to change7in the relative humidity similar to the ones seen during O3a. However WE PD1 calibrationstarted to experience changes during O3b up to 1 . . absolute timing of the PCal-induced motion has been measured to 110 ± µ s and takeninto account in the reconstruction of the gravitational wave signal.The future improvements foreseen for the PCal stability or for the PCal laser reflectedpower accuracy will help reducing the uncertainty of the online h ( t ) provided to data analysis.For the next observing runs, we can expect to reach and keep below 1% the uncertainty forthe PCal-induced displacement of the end mirrors. Acknowledgements
The authors gratefully acknowledge the University of Grenoble Alps Excellence Initiative(IDEX) and the United States National Science Foundation (NSF) for funding the D. Esteveztravel grant to LIGO Hanford Observatory (LHO). The authors gratefully acknowledge R.L. Savage and Y. Lecoeuche for their contribution to the lab measurements performed atLHO for the intercalibration of the LIGO and Virgo PCals. The authors also gratefullyacknowledge the Italian Istituto Nazionale di Fisica Nucleare (INFN), the French CentreNational de la Recherche Scientifique (CNRS) and the Foundation for Fundamental Researchon Matter supported by the Netherlands Organisation for Scientific Research, for theconstruction and operation of the Virgo detector and the creation and support of the EGOconsortium.
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