Towards optomechanical parametric instabilities prediction in ground-based gravitational wave detectors
David Cohen, Annalisa Allocca, Gilles Bogaert, Paola Puppo, Thibaut Jacqmin
OOptomechanical parametric instabilities simulation in Advanced Virgo
David E. Cohen, Annalisa Allocca, Gilles Bogaert, Paola Puppo, and Thibaut Jacqmin ∗ (Virgo Collaboration) Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France Universit`a di Napoli “Federico II”, Complesso Universitario di Monte S.Angelo, I-80126 Napoli, Italy Artemis, Universit´e Cˆote d’Azur, Observatoire Cˆote d’Azur, CNRS, F-06304 Nice, France INFN, Sezione di Roma, I-00185 Roma, Italy Laboratoire Kastler Brossel, Sorbonne Universit´e, CNRS,ENS-Universit´e PSL, Coll`ege de France, 75005 Paris, France (Dated: February 23, 2021)Increasing the laser power is essential to improve the sensitivity of interferometric gravitationalwave detectors. However, optomechanical parametric instabilities can set a limit to that power. It isof major importance to understand and characterize the many parameters and effects that influencethese instabilities. We perform simulations for the Advanced Virgo interferometer (O3 configura-tion). In particular we compute mechanical modes losses by combining both on-site measurementsand finite element analysis. We also study the influence on optical modes and parametric gains ofmirror finite size effects, and mirror deformations due to thermal absorption. We show that theseeffects play an important role if transverse optical modes of order higher than four are involved inthe instability process.
I. INTRODUCTION
In 2015, the LIGO-Virgo collaboration [1–4] detectedfor the first time gravitational waves preceding a binaryblack hole coalescence [5], thus pioneering gravitational-wave astronomy. Today many other gravitational waveshave been detected [6, 7]. These detections have pro-vided confirmation on the expected rate of binary blackhole (BBH) mergers [8], a better understanding of BBHspopulation [8, 9], a better limit to the mass of the gravi-ton [10], a first direct evidence of a link between bi-nary neutron star (BNS) mergers and short gamma-raybursts [11], a higher precision in constraining the Hubbleconstant [12], and a better understanding of BNS merg-ers [11]. Since the first detections, improvements per-formed on ground based detectors yielded better detectorsensitivities. Gravitational wave sources that are weakeror located further away can now be detected. Amongthe many improvements, increasing the light intensityin the interferometer arm cavities reduces the impact ofthe laser quantum phase noise, which is limiting the sen-sitivity in the high-frequency range. However, a laserpower increase can trigger a nonlinear optomechanicaleffect [13, 14], known as optomechanical parametric in-stability (OPI). This effect can jeopardize the interfer-ometer stable operation.During the Observing Run 1 (O1), LIGO experiencedan OPI for the first time [15]: after a few seconds, theinterferometer went out of lock, thus preventing furtherdata acquisition. In this letter, we present the modelsthat we used to compute the OPI gains in the configu-ration of Advanced Virgo. We then use these models tostudy many effects and parameters that can strongly af- ∗ Corresponding author: [email protected]
Laser BSDetectionphotodiodePR WIWE NI NE
FIG. 1. O3 Advanced Virgo’s configuration. NI and NE arerespectively the input and end-mirror of the North-arm, andWI and WE are respectively the input and end-mirror of theWest-arm. PR is the Power Recycling mirror, BS the inter-ferometer beam splitter. fect the amount of instabilities and their behavior in theAdvanced Virgo interferometer (O3 confiugration).In sec. II, a short introduction to the model used tocompute the parametric gains is given. In sec. III, thefinite element analysis (FEA) that was used in combi-nation with measurements to obtain mirror mechanicalmodes frequencies and quality factors is described. Insection IV, different models for optical modes are com-pared: the analytical solution of the paraxial equationfor purely spherical infinite size mirrors, and a brute forcenumerical simulation which includes finite size effects andarbitrary mirror surface shapes. In section V, we studythe influence on optical modes of a thermal effect relatedto a local temperature increase of the mirror surface dueto light absorption. Finally in section VI, we providean example of parametric gains that are obtained in the(O3) configuration. a r X i v : . [ a s t r o - ph . I M ] F e b II. OPTOMECHANICAL PARAMETRICINSTABILITY
In an optomechanical cavity like one arm of a gravi-tational wave detector, photons from the optical zerothorder mode can be coherently scattered to a higher ordertransverse optical mode if a mechanical mode that setsa mirror surface into motion has its frequency ω m / π equal to the frequency difference between the two opticalmodes (modulo the cavity free spectral range). This phe-nomenon can remove energy from the mechanical modeby annihilating phonons, and scattering photons from thezero order mode to the higher order transverse mode,thus damping the mirror motion [16]. Conversely, thisphenomenon can add energy to the mechanical modewith the reverse process, thus exciting the mechanicalmotion. In that case, an instability can prevent the in-terferometer stable operation [5, 13]. This instability hasa threshold: it starts to grow as soon as the resonant ex-citation of the mechanical mode by the radiation pressureforce overcomes mechanical losses.In the following, we use the approach developed byEvans et al. [17] to simulate this effect. In this frame-work, the whole interaction between the three impliedmodes (two optical modes and a mechanical mode) isseen as a classical feedback system. This modular ap-proach is well suited, since it can be adapted to manydifferent interferometer configurations with the same an-alytical formulas. The parametric gain of the mechanicalmode m is given by R m = 8 πQ m PM ω m cλ ∞ (cid:88) n =0 (cid:60) [ G n ] B m,n (1)where Q m is the quality factor of the mechanical mode m and ω m its frequency, P the arm-cavity optical power, λ the optical wavelength, M the mirror mass, c the velocityof light, G n is related to the scattered field optical gainof the n th optical mode and encapsulates the interferom-eter configuration. Finally, B m,n is the spatial overlapintegral between the three involved modes. A mechani-cal mode is amplified if R m > R m < R m > III. MECHANICAL SIMULATIONA. The spatial overlap parameter
The spatial overlap integral B n,m is defined [18] as B m,n = MM eff (cid:0)(cid:82) E ( (cid:126)r ) E n ( (cid:126)r ) µ m ⊥ ( (cid:126)r ) d(cid:126)r ⊥ (cid:1) (cid:82) | E | d(cid:126)r ⊥ (cid:82) | E n | d(cid:126)r ⊥ , (2)where M is the mirror mass and m eff the effective mass ofthe mechanical mode. The integral is performed over thetest mass surface (coating side). E ( (cid:126)r ) stands for the op-tical carrier amplitude and E n ( (cid:126)r ) for a transverse optical FIG. 2. Geometry used for the FEA, including the ears, theanchors and the magnets attached on the mirror rear face.The suspension wires are just for sketching but not included inthe simulation as they do not influence the modal frequencies mode amplitude labeled by the index n . As the interfer-ometer is sensitive to the test mass displacement alongthe optical axis, only the vertical displacement µ m ⊥ ( (cid:126)r ) isconsidered, where m is the mechanical mode index. Theeffective mass is related to the strain energy ρ e throughthe equation M eff ω m = (cid:82) ρ e (cid:126)r ⊥ , and effectively obtainedwith the formula M eff = M < µ ( (cid:126)r ) > = M V (cid:90) µ ( (cid:126)r ) d(cid:126)r, (3)where µ ( (cid:126)r ) stands for the test mass displacement.The mechanical modes were computed by means of fi-nite element analysis (FEA) developed for the actual in-put test mass (IM) of Advanced Virgo arm cavities [21].We have used the program Ansys ® Workbench TM . TheIM model includes the high-reflectivity (HR) coating ofthe front face, the flats and the bevels. Moreover theears and the anchors attached by silicate bonding tech-nique are included (see figure 2). In the FEA, the multi-layer optical coating is modelled as a solid 3D elementhaving the total thickness corresponding to the sum ofthe thicknesses of the high reflective and low reflective Advanced Virgo FEA parametersIM Coating a T a O High indexlayer overall thickness( t IM H ) 2080 nm SiO Low indexlayer overall thickness( t IM L ) 727 nmLoss angle ( φ CIM ) 1 . · − f . EM Coating a T a O High indexlayer overall thickness ( t EM H ) 3766 nm SiO Low indexlayer overall thickness ( t EM L ) 2109 nmLoss angle ( φ CEM ) 2 . · − f . TM SuprasilYoung modulus 72.251 GPaPoisson ratio 0.16649Density 2201 kg.m -3 Loss angle ( φ Suprasil ) 7 . · − f . Ear and anchors HCB [19]Young’s modulus 72.9 GPaPoisson ratio 0.17Density 2201 kg.m -3 Thickness 60 nmLoss angle ( φ HCB ) 0.1IM and EM propertiesMass 42 kgThickness 200 mmDiameter 350 mmFlats 50 mmTABLE I. Mechanical parameters used in the FEA aa For the FEA the multi-layer coating of the IM was replaced byone layer having the total thickness corresponding to the sumof the thicknesses of the high reflective and low reflectivematerials. The mechanical parameters used are the averagevalues of this layer [20] materials and mechanical parameters averaged over thethicknesses of the layers. Instead of 3D shell elements , wehave used 3D solid elements also for very thin materials,though more CPU time consuming, because they providethe shear deformations and energies, which are useful forgetting the mechanical losses associated to the modes.
B. FEA simulations results
The flats, the ears and also the anchors play an impor-tant role. In particular, since they break the cylindricalsymmetry, they lift degeneracies and increase the num-ber of distinct mode frequencies. In this paper we willdiscuss the results up to 70 kHz.To estimate the accuracy of the model, we have used aset of frequencies ( ν Meas ) measured on the North arm IM -8 10 -3 -6 10 -3 -4 10 -3 -2 10 -3
02 10 -3 -3 -3 -3 ∆ν/ν Frequency (Hz)
FIG. 3. Relative differences of measured frequencies withrespect to frequencies obtained with the FEA vs the FEAfrequencies. The standard deviation is 0 . · % up to 40kHz of an IM. Fig. 3 shows relative differences( ν Meas − ν FEA ) /ν FEA , versus the frequency of the FEA ν FEA . The standard deviation is 0.15%. v
Modal Frequencies [Hz] -9 -8 -7 -6 -5 FS + ITM coating + BondingFS + ITM coatingFS
FIG. 4. Loss angles obtained from the FEA of the InputTM. The computation ha been performed up to 70 kHz.
We have estimated the quality factors of the mechan-ical modes of the IM taking into account several kindsof losses: losses of the fused silica substrate, anchors andsupports of the magnets (loss angle φ FS ); coating losses(loss angle φ IMcoating ); losses of bonding layers used toattach the ears, the anchors and the magnets (loss angle φ Bonding ). The bonding layers, have a thickness of 60 nm,and are modeled as 3D solid elements. Coating losses ofthe IM and EM were recently measured [20]. Note thatall the parameters used are given in table I. Each losscontributor is related to the energy fraction stored in thelossy part and to the material loss angle, through the
Modal Frequencies [Hz] Q FIG. 5. Quality factors of the mechanical modes up to 70kHz relationships φ Bonding · E tot = φ HCB · E bonds φ IMcoating · E tot = φ CIM · E CIM φ FS · E tot = φ Suprasil · E FS . (4)The overall loss angle for the IM is obtained by summingup all contributors: φ IM = φ Bonding + φ IMcoating + φ FS .The mechanical quality factor of the IM modes thenwrites Q m = 1 /φ IM .Fig. 4 shows the frequency dependence of the FS sub-strate loss and the effect of adding the optical coatingand the bonding layers. The influence of the bondingterm φ Bonding is strongly mode shape dependent throughthe deformation of the ear and anchor bulks and it isnot negligible. In fact, its contribution to Q m is domi-nant. For this reason, from a set of Q measurements itis possible to infer the value of φ HCB by using the energyfractions calculated with the FEA.Fig. 5 shows the Q m of the IM mass computed byfitting the loss angle φ HCB by using the first set of 5modes of the IM of the north arm and supposing thatit is does not vary with the frequency. At frequencieshigher than 10 kHz, the bondings have a strong dampingeffect, though they have a negligible effect on the ther-mal noise of the IM. This is a very important result forthe parametric gains computation and consequently foridentification of the unstable modes.
IV. TRANSVERSE OPTICAL MODES IN ARMCAVITIES
Hermite-Gauss modes (HGM) are solutions of theparaxial wave equation for infinite-sized spherical mir-rors. This mode basis was used in [17] to compute theparametric gain for the LIGO interferometer. It is fastto implement as the mode shapes are provided by ana-lytical formulas. However, it restricts the mirror model to a purely spherical shape of infinite size. In particu-lar, it does not include the effects of the deviations fromthe spherical shape due to fabrication imperfections orthermal effects. Finally, it does not take account for fi-nite size effects such as diffraction losses, which must beestimated separately.We have computed another set of optical modes thatare obtained from a numerical resolution of the paraxialequation with finite-sized mirrors [22]. This mode basiswill be referred to as ‘finite-sized mirror modes (FSMM)’.Contrary to HGM, FSMM are obtained directly withdiffraction losses. Moreover, mirror shapes can be chosenarbitrary, which enables one to introduce any deforma-tion of the mirrors due to thermal effects or fabricationimperfections. Note, that in this work, we did not in-clude fabrication imperfections, which effects will be thesubject of future work.In the following, we analyze the differences in Gouyphase (or frequency), diffraction loss, and mode shape,and between the HGM and FSMM basis set.
A. Gouy phases
The Gouy phases of arm cavity modes set the opticalresonant frequencies, and, thus, the OPI resonance con-dition ω m = δω , where δω is the difference in frequencybetween the zero order mode and the higher order trans-verse optical mode. In the case of HGM, the Gouy phaseof the mode of linear index n is given by φ G n = O n φ G (5)where O n the order of HGM(n), and φ G is the Gouyphase of the lowest order mode HGM(1) (usually referredto as TEM in the literature) given by φ G = arccos ( −√ g g ) . (6)Here, g < g < φ G (cid:39) .
74 rad. and can be tuned by small vari-ations of the mirror radii of curvature. Fig. 6 shows thedifference between HGM’s and FSMM’s Gouy phases, ex-pressed in units of free spectral range on the left verticalaxis and in units of cavity linewidth on the vertical rightaxis. Note that the Gouy phases have been wrappedwithin an interval of length π , which allows to fold allthe modes within a single free spectral range. The greenline splits the graph into two regions: in the above re-gion, the deviation is more than half a cavity linewidth,and we expect the model choice to have an impact on theOPI gain, whereas in the bottom part the impact shouldbe negligible. Thus, the critical order is 7. B. Diffraction losses
Diffraction losses stem from the finite size of the cav-ity mirrors. They are a key parameter to compute the
FSMM and HGM index -8 -7 -6 -5 -4 -3 -2 -1 F r equen cy d i ff e r en c e ( F S R un i t ) -4 -2 F r equen cy d i ff e r en c e ( li ne w i d t h un i t )
01 2 3 4 5 6 7 8 9 10 11
FIG. 6. Difference between HGM’s and FSMM’s Gouy phase,expressed in units of free spectral range on the left verticalaxis, and in units of cavity linewidth on the right verticalaxis. The vertical red dashed lines highlight the mode orders,which appear above the upper horizontal axis. The horizontallower axis shows the optical mode index (modes are sorted byincreasing energies). The green horizontal line has a verticalcoordinate of 0.5 on the right axis. parametric gain, since they contribute to the opticallinewidth (together with material absorption losses, scat-tering losses, and mirror transmittance). Since low-ordermodes have most of the energy concentrated at the centerof the mirror, their diffraction losses are small, whereashigh-order modes spread over a larger surface and showhigher diffraction losses. Thus, in general, high-ordermodes are less likely to contribute to a PI. However,note that counter-intuitively, a loss increase can some-times lead to a higher parametric gain, as explained inmore details in section VI B.Diffraction losses for HGM are estimated, like in [17],by evaluating the ratio between the total light flux withinthe coating radius of a mirror and the total flux incidenton the mirror. Figure 7 shows diffraction losses for bothsets of modes. It shows that with this rough estima-tion method, besides the HGM(1), all HGM have theirdiffraction losses underestimated. However, note that thetotal losses (input mirror transmittance plus diffractionlosses) of low-order modes are dominated by the inputmirror transmittance (green line on Fig. 7). Thus, thetotal losses obtained with the two methods start to differby more than 10% around mode order 5.
C. Mode amplitudes
Optical mode amplitudes are used to compute thethree mode spatial overlap coefficient B mn of Eq. 1.Thus, they also directly affect the OPI gain. In order FSMM and HGM index -2 D i ff r a c t i on l o ss ( pp m ) FIG. 7. Diffraction losses obtained for FSMM (cross) andestimated for HGM (circles). The vertical red dashed linesunderline the mode orders, which appear in the upper hori-zontal axis. The horizontal lower axis shows the optical modeindex (modes are sorted by increasing energies). The greenline shows the input mirror transmittance. to compare the mode amplitudes of FSMM and HGM,we decompose the vectors of one basis set onto the otherby using the decomposition coefficient c ij of any FSMM(index i ) with any HGM (index j ): c ij = (cid:90) (cid:90) S d x d y u ∗ i ( x, y ) v j ( x, y ) , (7)where i and j are modes integer indices, u i (resp. v j ) arethe FSMM (resp. HGM) mode amplitudes, and S is themirror coating surface. Note that the transverse profileof a FSMM is constrained on a disk (mirror coating),whereas for HGM the transverse profile is distributedover the whole plane, such that a linear superpositionof FSMM will never exactly match a HGM, and a truetransformation matrix between the two basis set cannotrigorously be obtained [23]. In Fig. 8(a), we represent | c ij | for i = 2, and j ∈ { , , ..., } . We find thatFSMM(2) is a linear combination of the two order oneHGM which are the HGM(2) and the HGM(3). We findthat this is true for orders below 7. Conversely, as shownin Fig. 8(b), the higher order FSMM(36) mode (shown inthe inset of Fig. 8(b)) cannot be decomposed on a singleorder of HGM. In the presented case, it is a mixture oforder 7, 9, 11, and many other higher odd orders thatare not shown on the figure. D. Conclusion
This study shows that, in the absence of mirror de-formation, the HGM basis does not deviate significantlyfrom the FSMM basis for modes of order lower than 6.
HGM index | c ij | (a) TEM HGM index | c ij | (b) FIG. 8. (a) | c ij | terms of the decomposition of the FSMM(2)on the HGM basis: the FSMM(2) is a linear combination ofthe HGM(2) and the HGM(3). (b) | c ij | terms of the decom-position of the FSMM(36) on the HGM basis. For order 6 and higher, the more resource consumingFSMM basis should lead to significantly different resultsfor OPI gains. In section VI, we compare the OPI gainsobtained for the Advanced Virgo O3 configuration, withHGM and FSMM basis set.
V. THERMAL EFFECTS
The laser energy is partially absorbed both by coatingsand in the bulk of mirrors. This causes a temperaturegradient, which originates two effects. First, a gradientof refractive index in the bulk of input mirrors modifiesthe mode matching condition, but affects neither the cav-ity linewidth nor the mode frequencies. Second, a defor-mation of the mirror surface, which modifies the modeshapes and frequencies. In this part, we evaluate the impact on this second effect on the properties of cavitymodes by comparing FSMM obtained for purely spheri-cal mirrors with FSMM obtained for thermally deformedmirrors. -15 -10 -5 0 5 10 15
Position (cm) -8 -7 -6 -5 A m p li t ude ( m ) (a) FSMM index F r equen cy d i ff e r en c e ( F S R un i t ) (b) F r equen cy d i ff e r en c e ( li ne w i d t h un i t )
01 2 3 4 5 6 7 8 9 10 11
FIG. 9. (a) Input mirror profiles with (dashed red line) andwithout thermal effect (blue solid line). (b) Frequency differ-ence between FSMM modes with and without thermal effect,expressed in unit of FSR on the left axis, and in units oflinewidth on the right axis. The green horizontal line is atvertical position 0.5 on the right axis.
The deformation profile is obtained by solving the lin-ear thermoelastic equations [24]. Figure 9(a) shows thepurely spherical and thermally deformed profiles of anAdvanced Virgo input mirror, for an intracavity powerof 300 kW. We fitted the central part of the deformedmirror to extract a radius of curvature. The results aregiven in the following table:No thermal effect With thermal effectInput mirror 1424.6 m 1432.1 mEnd mirror 1695 m 1702.3 mHowever, note that the mirror is not spherical any-more and the result of the fit is only valid in the cen-ter. In order to evaluate the incidence of this effect onthe optical cavity parameters, we computed the FSMMwith and without thermal effect on the two cavity mir-rors. Fig. 9(b) shows the frequency differences betweenthe two situations. We see that optical modes acquire asignificantly different Gouy phase even for very low modeorders. We checked that losses and mode amplitudes areaffected only for orders higher than 7, such that the fre-quency shift is the main effect. In section VI, we studythe impact of this phenomenon on OPI gains.
VI. PARAMETRIC GAIN COMPUTATIONA. Validation: comparison with the Finessesoftware R m Frequency (kHz) -4-2024 R e l a t i v e e rr o r ( % ) FIG. 10. (a) Parametric gain obtained with the Finesse soft-ware for one mechanical mode (12.552 kHz) on mirror NE(blue) and WE (red). (b) Relative difference between (a) andthe parametric gain R m obtained with Eq. (1) using FSMM(solid lines) and HGM (dashed lines). The OPI gains of all mechanical modes within the [5.7,70.7] kHz range were computed using Eq. (1) using boththe HGM and FSMM basis set. In order to validate thismethod, we compared our results with the one obtainedwith the Finesse software [25, 26]. The OPI gain ob-tained with the Finesse software for one mechanical modeand two arm cavity mirrors is shown in Fig. 10(a). InFig. 10(b), we plot the relative difference with the OPIgain obtained with the Finesse software and with Eq. (1)using FSMM and HGM. We observe a difference of afew percent at maximum. Note that the slight asym-metry between the blue and red curves stems from thesmall parameter difference between the two arms cavi-ties. This comparison has been performed with many other mechanical modes and showed similar results. Notethat using Eq. (1) is much faster than using the Finessesoftware, and that computing the results of the follow-ing figures would not have been possible in a reasonableamount of time. Therefore, in the following we use onlyEq. (1).
B. Effect of optical losses on the OPI gain
In this section we demonstrate a counter-intuitive ef-fect of optical losses on the OPI gains. Intuitively, ifoptical losses increase, the OPI gains get lower since theoptical linewidth also increases. Here we show that if theOPI resonance condition is not exactly fulfilled, broad-ening the optical mode response can increase the gainsuch that counter intuitively, the gain variation does notvary monotonously with the diffraction loss. This is bestshown on Fig.11(a), where the OPI gain of a mechani-cal mode is plotted against optical diffraction losses ofthe main optical contributor. In this example, the gainfirst increases from around 0.04 below 10 ppm to 0.1at 2 × ppm, before decreasing at higher loss values,as expected. This appears also in Fig. 11(b), where theoptical gain G n of the main optical contributor to theOPI gain of the mechanical mode of Fig. 11(a) is repre-sented as a function of the mechanical mode frequency,for two different values of diffraction losses. At low losses,the two resonance peaks are well separated, such thatthere is a minimum in between (black arrow). A loss in-crease from 0 to 21540 ppm (red bullet at maximum ofFig. 11(a)) broadens the peaks an lead to the red curvewhich has no minimum anymore, and which shows highervalues in a whole frequency region (gray shaded area onFig. 11(b)). Finally, if the losses were increased further,the red curve would start lowering and the gray shadedarea would vanish. C. Impact of optical mode basis set on OPI gain
In this paragraph, we study the impact of the modelused to compute the optical modes. We compare the OPIgains obtained with the HGM and FSMM basis. As ex-pected, we find that there is only a marginal differencebetween the two models if optical modes of order below 5are involved. In Fig. 12, we plot the gain of a mechanicalmode versus its frequency using the two optical mode ba-sis set. This mode has been chosen for the main opticalcontributor to the OPI gain is an order 6 optical mode.There is a factor 3 between the two gain maxima andthe two peaks are shifted by around 100 Hz, which cor-responds to the optical linewidth. This is in agreementwith the conclusions of Sec. IV. Diffraction losses (ppm) R m (a) Mechanical mode: 12.551 kHz Main optical contributor12.5 12.55 12.6 12.65 12.7 12.75 12.8
Mechanical mode frequency (kHz) G n (b) FIG. 11. (a) Parametric gain R m of a mechanical mode offrequency 12.5551 kHz while varying artificially the diffractionlosses of optical modes. (b) Optical gain G n of a FSMM highorder mode versus the mechanical mode frequency, which isartificially varied around the resonance condition δω = ω m (with δω = 12.5551 kHz). The blue line is for null diffractionlosses, and the red one is for 21540 ppm (red bullet at themaximum of the curve in (a)). The arrow point the minimumin between the two resonance on the blue curve, where thegain increase is maximum. The gray shaded area highlightsa frequency at which G n is higher when the diffraction loss ishigher. D. OPI gain computation in the O3 configuration
The simulations have been performed for the AdvancedVirgo configuration corresponding to that of O3. Theparameters for such a configuration are shown in Ta-ble II. Measured parameters have been included ratherthan nominal values when they were available. Only theoptical input power has been set to the nominal value of50 W, which is the maximum value that would have been
Frequency (kHz) -0.0500.050.10.150.20.250.30.35 R m Mechanical mode: 33.750 kHz
FIG. 12. Parametric gain R m of the 33.750 kHz mechanicalmode (see inset) while artificially varying the mechanical fre-quency, using the two different optical mode basis (red: HGM,blue: FSMM). The dashed lines point the maxima of the twocurves, and emphasize the height and frequency change.Arm lengths 2999.8 mTransmittance NI 13770 ppmTransmittance NE 4.4 ppmTransmittance WI 13750 ppmTransmittance WE 4.3 ppmTransmittance PR 48400 ppmRound trip loss 75 ppmDistance from BS to NI 6.0167 mDistance from BS to WI 5.7856 mDistance from BS to PR 6.0513 mLaser wavelength 1064 nmGouy phase of PR cavity 1.8 mradTABLE II. Advanced Virgo O3 optical parameters possibly used during O3 (the value effectively reached inO3 being too small to trigger any instability in the rangeof mechanical mode frequencies simulated in this work).The corresponding arm-cavity power is around 300 kW.The parameters used are listed in table II.To account for optical mode frequency uncertainties,we present OPI data in two-dimensional plots (see forinstance Fig. 13(a) and (b)), where the end mirrors radiiof curvature NE and WE are scanned. The color codeindicates the gain value at each interferometer workingpoint. This choice is also related to the fact that the en-visioned OPI mitigation technique relies on ring heatersable to tune the end mirrors radii of curvatures [27]. Inthe following, we show two sets of results. Each figure isthe result of the same OPI calculation but using a dif-ferent set of optical modes. Fig. 13(a, b, c) shows theresults for FSMM, Fig. 13(d, e, f) for FSMM includingthermal effect due to coating absorption (see sec. V). NE A B C D EF/CG (a)(c) (b) (d)(f) (e)ABC DEFG
FIG. 13. (a) Parametric gain R m versus radii of curvature of NE and WE, using FSMM without thermal effect, for an inputpower of 50 W. The gray-shaded scale highlights the parametric gains lower than 1, while the colorful scale highlights theinstabilities ( R m > α contributes for 93 % of A and for 77 % of B, β contributes for 6 % of A andfor 15 % of B, γ contributes for 89 % of C, and δ for 10 % of C. (c) The red solid line is the number of unstable modes in theradius of curvature range of (a), with respect to the optical input power (top) or intracavity power (bottom). They becomeunstable from an optical input power of respectively 9 W (mode A), 26 W (mode B), and 27 W (mode C). The blue curve isthe ratio of the area free of instability S Rm> to the total area S tot in Fig. 13 (a resp. b), versus the optical power. The figures(d) resp. (e) resp. (f) are the same than (a) resp. (b) resp. (c), but taking into account the thermal effect. The five unstablemechanical modes are labeled D, E, F, G, and C (same mode than without thermal effect). Their frequencies are respectively66.888 kHz, 66.912 kHz, 61.216 kHz, 61.231 kHz, and 66.784 kHz. (cid:15) contributes for 84 % of D and 24 % of E, ζ contributes for16 % of D and 76 % of E, η contributes for 100 % of F and G, and θ contributes for 97 % of C. They become unstable froman optical input power of respectively 4.6 W (mode D), 35 W (mode E), 42 W (mode F), 51 W (mode G), and 57 W (modeC). The green dashed lines show the input power reached in O3 (28 W), and the nominal power (50 W) corresponding to (a)and (b). and WE are scanned over a five meters range, which iswithin reach of the mirror ring heater system. In eachOPI plot, the color code is chosen such that the gray scaleis for gains lower than 1 (no instability), and the colorscale is for R m >
1. The involved mechanical modes areindicated in the inset, and main optical modes contribu- tors are shown below each OPI plot. Note that the resultobtained with HGM are indistinguishable from that ofFig. 13(a), such that we did not include the correspond-ing figure. Indeed, only low order optical modes are in-volved here, such that HGM and FSMM give the sameresult. Fig. 13(b resp. d) shows the unstable mechanical0modes of Fig. 13(a resp. b) on the first line. The secondline shows the involved optical modes. In Fig. 13(c resp.f), we plot the number of unstable modes in the rangeof Fig. 13(a resp. b) versus the optical power for FSMMwithout (resp. with) thermal effect. Modes that ring ondifferent mirrors are counted only once. The blue curvesrepresents the ratio of the area free of instability S Rm> to the total area S tot in Fig. 13 (a resp. b), versus theoptical power. This ratio quantifies how difficult it is toescape an unstable area within the accessible radii of cur-vature range. The green vertical lines on Fig. 13(c andf) point the nominal power of O3 (50 W) and the powerthat was effectively reached (28 W).These results show that an OPI involving mechanicalmodes with frequencies below 70 kHz could have beenobserved at the nominal power of O3, although it wouldbe easily escaped with end-mirrors ring heaters since S Rm> / S tot (cid:39) . VII. CONCLUSION
In this letter, we have presented OPI gain simulationsin the Advanced Virgo configuration of O3. Compared toprevious work [17], we have used deeper physical mod-eling, including a very detailed description of mechani- cal modes and optical modes. In particular, arm-cavitymirrors mechanical modes frequencies and quality factorswere evaluated by combining measurements with FEA,including most of the mirrors details. We also have stud-ied different models and techniques to compute opticalmodes of arm-cavities. We have shown that up to or-der 4 (included), analytical formulas for Hermite-Gaussmodes are sufficient to predict accurate OPI gains. How-ever, if higher order optical modes are involved, mirrorsfinite size effects must be accounted for. Finally, we haveshown that the mirror deformation stemming from thelaser absorption in mirror coatings plays an importantrole and must be included in the OPI simulation. Thesesimulations pave the way towards precise optomechanicalinstability predictions for the next observing run (O4).
ACKNOWLEDGMENTS
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