Parametric instabilities in the LCGT arm cavity
K Yamamoto, T Uchiyama, S Miyoki, M Ohashi, K Kuroda, K Numata
aa r X i v : . [ g r- q c ] M a y Parametric instabilities in the LCGT arm cavity
K Yamamoto , T Uchiyama , S Miyoki , M Ohashi , K Kuroda ,K Numata Institute for Cosmic Ray Research, The University of Tokyo, Kashiwa, Chiba 277-8582,Japan NASA Goddard Space Flight Center, CRESST, Code663, Greenbelt, MD 20771, U.S.A.E-mail: [email protected]
Abstract.
We evaluated the parametric instabilities of LCGT (Japanese interferometricgravitational wave detector project) arm cavity. The number of unstable modes of LCGTis 10-times smaller than that of Advanced LIGO (U.S.A.). Since the strength of the instabilitiesof LCGT depends on the mirror curvature more weakly than that of Advanced LIGO, therequirement of the mirror curvature accuracy is easier to be achieved. The difference in theparametric instabilities between LCGT and Advanced LIGO is because of the thermal noisereduction methods (LCGT, cooling sapphire mirrors; Advanced LIGO, fused silica mirrors withlarger laser beams), which are the main strategies of the projects. Elastic Q reduction bythe barrel surface (0.2 mm thickness Ta O ) coating is effective to suppress instabilities in theLCGT arm cavity. Therefore, the cryogenic interferometer is a smart solution for the parametricinstabilities in addition to thermal noise and thermal lensing.
1. Introduction
Observations using several interferometric gravitational wave detectors (LIGO [1], Virgo [2],GEO [3], TAMA [4]) on the ground are presently in progress. In order to construct detectorswith better sensitivity, future (second generation) projects have been proposed: Advanced LIGO(U.S.A.) [5] and LCGT (Japan) [6]. These projects have km-scale Fabry-Perot cavities. Theoptical transversal mode spacing in these long cavities and the intervals of the elastic modes of themirrors are on the order of 10 kHz. When the optical transversal mode spacing are comparablewith the intervals of the elastic modes, parametric instabilities become a problem in the stableoperation of the interferometer [7]. Small thermally driven elastic vibration modulates the lightand excites the transverse modes of the cavity. These excited optical modes apply modulatedradiation pressure on the mirrors. This makes the amplitude of the elastic modes larger. Atlast, the elastic modes and optical modes, except for TEM00, oscillate largely.The formula of the parametric instability (without power recycling, resonant sidebandextraction, and anti-Stokes modes) is derived in Ref. [7]. If a parameter, R , of an elasticmode is larger than unity, that mode is unstable. The formula of R is R = X optical mode P Q m Q o M cLω m2 Λ o ω /δ o2 , (1)where P, Q m , Q o , M, c, L, ω m , ∆ ω , and δ o are the optical power in the cavity, the Q-values ofthe elastic and optical modes, the mass of the mirror, the speed of light, the cavity length,he angular frequency of the elastic mode, the angular frequency differences between the elasticand optical modes, and the half-width angular frequency of the optical mode, respectively. Thevalue Λ o represents the spatial overlap between the optical and elastic modes. If the shapesof the optical and elastic modes are similar, Λ o is on the order of unity. If the shapes are notsimilar, Λ o is almost zero. When the shapes and frequencies of the optical and elastic modesare similar (Λ o ∼ , ∆ ω ∼ R will become several thousand in future projects [8]. Theseparametric instabilities are a serious problem in Advanced LIGO [9, 10]. The instabilities of theLCGT interferometer have never been considered. We evaluated the instabilities of the LCGTarm cavity. In Sec. 2, we introduce the calculation results for the parametric instabilities inLCGT and compare them with those in Advanced LIGO. In order to calculate ω m and Λ o forthe instability evaluation, we used ANSYS, which is a software application for a finite-elementmethod [11]. In Sec. 3, the difference in the parametric instabilities between Advanced LIGOand LCGT is discussed. In Sec. 4, how to suppress the instabilities of the LCGT arm cavity isconsidered. Sec. 5 and Sec. 6 are devoted to future work and a summary, respectively.
2. Results of calculation
Table 1 gives the specifications of Advanced LIGO in Refs. [9, 10, 12] (after these references, thespecifications of Advanced LIGO were changed slightly) and LCGT [13] (the exact values of theLCGT mirror curvature are not fixed. The curvature given in Table 1 is only a candidate). Theimportant differences between Advanced LIGO and LCGT are in the mirror curvature radius,beam radius, mirror material and temperature.
Table 1.
Specification of Advanced LIGO [9, 10, 12] and LCGT [13].Advanced LIGO LCGTLaser beam profile Gaussian GaussianWavelength 1064 nm 1064 nmCavity length 4000 m 3000 mFront mirror curvature radius 2076 m 7114 mEnd mirror curvature radius 2076 m 7114 mBeam radius at the mirrors 60 mm 35 mmPower in a cavity 0.83 MW 0.41 MWMirror material Fused silica SapphireMirror mass 40 kg 30 kgMirror temperature 300 K 20 K
We briefly overview an estimation of the instabilities in Advanced LIGO by a group at theUniversity of Western Australia [9, 10] for an easier comparison. They investigated what happenswhen the curvature of a mirror is changed. The curvature of the other mirror is the defaultvalue given in Table 1. Figure 1(c) of Ref. [10] shows the curvature dependence of the unstablemode number. The crosses in this figure represent the cavities, which consist of the fused-silicamirrors. The number of unstable modes is between 20 and 60. Figure 5 of Ref. [9] shows thatthe maximum of R in the various elastic modes strongly depends on the mirror curvature. Evena shift of only a few meters in the mirror curvature causes a drastic change of the maximum R .he requirement of the accuracy in the mirror curvature in Advanced LIGO is difficult to beachieved. We investigated the parametric instabilities of the LCGT arm cavity in the same manner asthat of the University of Western Australia. Figure 1 shows the mirror curvature dependenceof the unstable mode number in the LCGT arm cavity. The number is only 2 ∼
4, whichis 10-times smaller than that of Advanced LIGO. Figure 2 shows that the mirror curvaturedependence of the maximum R is weaker than that of Advanced LIGO. The maximum R is notchanged drastically by a shift of a few meters in the mirror curvature. It is easier to satisfythe requirement of the mirror curvature in LCGT. It must be noticed that the power recyclingwas not taken into account in our calculation. The evaluation of Advanced LIGO included thepower recycling effect. The difference between the maximums of R with and without the powerrecycling is shown in fig. 2 of Ref. [10]. Although the peaks of the maximum R become higherowing to the power recycling, this is not a serious effect on the discussion about the differencebetween Advanced LIGO and LCGT in the next section. N u m b er o f un s t a b l e m o d e s Figure 1.
Number of unstable modes in theLCGT arm cavity. The horizontal axis is thecurvature radius of a mirror. The curvature ofthe other mirror is the default value given inTable 1. This graph corresponds to fig. 1(c)of Ref. [10] for Advanced LIGO. M ax i m u m o f R Figure 2.
Maximum of R in the variouselastic modes in the LCGT arm cavity. Thehorizontal axis is the curvature radius of amirror. The curvature of the other mirror isthe default value given in Table 1. This graphcorresponds to fig. 5 of Ref. [9] for AdvancedLIGO.
3. Discussion
Our investigation revealed that there is the large difference in the parametric instabilities betweenAdvanced LIGO and LCGT. We discuss the reasons in this section.
The difference in the unstable mode numbers originates from the mode frequency densitydifference. If both of the optical and elastic mode densities are large, the optical modefrequencies often coincide with the elastic mode frequencies. The elastic mode density is inverselyproportional to the cube of the sound velocity in the material. The sound velocities of the fusedsilica (Advanced LIGO) and sapphire (LCGT) are about 6 km/sec and 10 km/sec, respectively.The elastic mode density of LCGT is 5-times smaller. In Advanced LIGO, there are 7 opticalransverse modes in a free spectrum range [9]. On the contrary, there are only 3 modes in theLCGT arm cavity. The optical mode density of LCGT is 2-times smaller. The larger opticalmode density of Advanced LIGO stems from a larger beam radius adapted for suppressing themirror thermal noise [5]. In LCGT, since the mirrors are cooled in order to reduce the thermalnoise [6], larger beams are not necessary. As a result, the product of the elastic and optical modedensities of LCGT becomes 10-times smaller. In fact, the unstable mode number calculated byus for LCGT (2 ∼ ∼
60 in fig. 1(c) of Ref. [10]).
In LCGT, the maximum value of R depends on the mirror curvature more weakly. This impliesthat the curvature dependence of the optical mode frequencies is weaker, because R is a functionof the optical mode frequencies. We calculated how the curvature variation affects the n -thoptical transverse mode. The results were 15 n Hz/m in Advanced LIGO and 0.58 n Hz/m inLCGT. LCGT shows a 30-times weaker dependence due to the larger optical transversal modespacing, which stems from the smaller beam radius.
The difference in the parametric instabilities between Advanced LIGO and LCGT is caused bythose of the beam radii (Advanced LIGO, 60 mm; LCGT, 35 mm) and the mirror materials(Advanced LIGO, fused silica; LCGT, sapphire). These differences mostly originate from thatof the thermal noise-reduction methods (Advanced LIGO, fused silica mirrors with larger laserbeams; LCGT, cooling sapphire mirrors), which are the main strategies of the projects [5, 6].The cryogenic interferometer has an advantage in the parametric instabilities (less unstablemode number and weaker dependence on the mirror curvature) in addition to the small thermalnoise [14, 15, 16] and negligible thermal lensing [17].
4. Instability suppression for LCGT arm cavity
We showed that the unstable modes of LCGT are less than those of Advanced LIGO. Still, theLCGT arm cavity is unstable because there always exist unstable modes, as shown in fig. 1. Wemust consider how to suppress the instabilities. The three methods for instability suppression inAdvanced LIGO are being studied at the University of Western Australia [10, 18]. We checkedwhether these three methods (thermal tuning method, feedback control, Q reduction of elasticmodes) are appropriate for LCGT (the tranquilizer cavity [19] is one of the other methods.However, this is difficult).
In the thermal tuning method [10], a part of the mirror is heated for curvature control. Since R depends on the curvature, the suppression of R should be possible by this manner. However,this method is not useful in LCGT. First of all, R weakly depends on the mirror curvature, asshown in fig. 2 in the case of LCGT. Second, owing to the small thermal expansion and highthermal conductivity [17] of cold sapphire, the mirror curvature would not change effectively,even if we apply heat to the mirror. It is possible to control the light or the mirror so that the parametric instabilities would beactively suppressed [10]. If the number of unstable modes is smaller, feedback control is easier.However, these are more difficult (active) methods than Q reduction (passive method) of theelastic modes. .3. Q reduction of elastic modes
This is a useful method [18] for LCGT. The value of R is proportional to the Q-value of the elasticmode, Q m , as shown in Eq. (1). The Q-values of sapphire are about 10 [14]. The maximum R of LCGT is several hundreds at most, as shown in fig. 2. If the Q-values of the LCGT mirrorsbecome 10 , almost all modes become stable. Since the mechanical loss concentrated far fromthe beam spot has a small contribution to the thermal noise [20, 21], we should be able to applyadditional loss on a barrel surface, as in fig. 3, without sacrificing the thermal noise [18, 22].Moreover, it is possible to introduce a 15-times larger loss in the LCGT mirror than that inAdvanced LIGO mirror, because the LCGT mirrors are cooled (20 K) [6]. We concluded thatthe thermal noise of the barrel surface loss is comparable with that of the reflective-coating loss[16, 21], when the LCGT mirror Q-values become 10 owing to the additional loss. Since thereflective-coating thermal noise is smaller than the goal sensitivity of LCGT [16], this barrelsurface loss is not a serious problem. Laser beam Additional lossfor dampingReflective coating
Figure 3.
Loss on the barrel surface.Although this loss decreases the elastic Q-values of the mirror, Q m , it has onlya small contribution to the thermal noise[20, 21]. Thus, this loss suppresses theparametric instabilities without an increase ofthe thermal noise [18, 22].We are able to introduce loss on the barrel surface by coating Ta O , which is a popularmaterial for the reflective coating of the mirror. Our recent measurement [16] proved that theloss angle of the SiO /Ta O coating is (4 ∼ × − between 4 K and 300 K. Since the lossof this coating is dominated by that of Ta O [23], the loss angle of Ta O is (8 ∼ × − .If the barrel loss dominates the mirror Q, it would be expressed as [21]1 Q m ∼ E Ta O E sapphire dR φ, (2)where E Ta O , E sapphire , d, R, φ are the Young’s moduli of Ta O and the sapphire, the thicknessof the Ta O layer, the mirror radius and the loss angle of Ta O , respectively. These valuesare summarized in Table 2 [16]. In order to make the Q-values, Q m , 10 , the Ta O coatingthickness, d , must be 0.2 mm. Table 2.
Specification of the coating [16].Young’s modulus of the Ta O ( E Ta O ) 1 . × PaYoung’s modulus of the sapphire ( E sapphire ) 4 . × PaMirror radius ( R ) 12.5 cmLoss angle of Ta O ( φ ) 10 − . Future work In our calculations for this article, we took only the elastic modes below 100 kHz and the firstthree transverse optical modes into account. We were calculating higher elastic and opticalmodes. Our preliminary result suggests that there are unstable higher modes. However, theelastic Q reduction would work more effectively because the typical R still seems to be small.We must evaluate the Q reduction technique more carefully for instability suppression. Theeffects of power recycling [8], resonant sideband extraction [24, 25] and the anti-Stokes modes[26] must be evaluated as well.
6. Summary
We evaluated the parametric instabilities of LCGT and compared them with those of AdvancedLIGO [9, 10]. The number of unstable elastic modes in LCGT is 10-times smaller. Since thestrength of the parametric instabilities in LCGT more weakly depends on the mirror curvature,the requirement of the accuracy in the mirror curvature of LCGT is easier to satisfy. Thesedifferences in the parametric instabilities between LCGT and Advanced LIGO are made bythose of the laser beam sizes and mirror materials, which mostly stem from the thermal-noisesuppression strategies in both projects (LCGT, cooling sapphire mirrors; Advanced LIGO, fusedsilica mirrors with larger laser beams) [5, 6]. The elastic Q reduction [18] by the barrel surface(0.2 mm thickness Ta O ) coating is effective for instability suppression in the LCGT arm cavity.Thus, the cryogenic interferometer is a smart solution for the parametric instabilities in additionto the thermal noise [14, 15, 16] and thermal lensing [17]. Acknowledgments
We are grateful to M Ando for information about candidates of the LCGT mirror curvatureradius.
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