Elsevier

Physics Letters A

Volume 370, Issues 3–4, 22 October 2007, Pages 177-183
Physics Letters A

Parametric instability in GEO 600 interferometer

https://doi.org/10.1016/j.physleta.2007.05.063Get rights and content

Abstract

We present analysis of undesirable effect of parametric instability in signal recycled GEO 600 interferometer. The basis for this effect is provided by excitation of additional (Stokes) optical mode, having frequency ω1, and mirror elastic mode, having frequency ωm, when the optical energy stored in the main FP cavity mode, having frequency ω0, exceeds a certain threshold and detuning Δ=ω0ω1ωm is small. We discuss the potential of observing parametric instability and its precursors in GEO 600 interferometer. This approach provides the best option to get familiar with this phenomenon, to develop experimental methods to depress it and to test the effectiveness of these methods in situ.

Introduction

The full scale operational terrestrial interferometric gravitational wave antennae LIGO have sensitivity, expressed in terms of the metric perturbation amplitude, approximately 3 times better than the planned level of h1×10−21 [1], [2] in 100 Hz bandwidth (see the current sensitivity curve in [3]). In advanced LIGO (to be approximately realized in 2012), after improving noise of test masses (mirrors of a 4 km long optical Fabry–Perot (FP) cavities) and increasing the optical power circulating inside the resonator the sensitivity is expected to reach the value of h1×10−22 [4], [5]. GEO 600 interferometer has absolute sensitivity less than LIGO, however, regarding to its armlength (1200 m, in LIGO 4 km) and circulating power (also lower than in LIGO) sensitivity of GEO 600 is an impressive one (see [6], [7]). In particular, GEO 600 has participated and continues to participate (with LIGO and others) in scientific run S5 aimed to register gravitational waves. In addition GEO 600 plays an important role as a testing area for different kinds of new technologies to be applied later for LIGO, e.g., the signal recycling configuration, monolithic fused silica suspension, usage of electro-static actuators, first demonstration of thermal lensing compensation and several others.

The undesirable effect of parametric instability in Fabry–Perot cavity, which may cause a substantial decrease in antennae sensitivity or even antenna malfunction, was examined in [8]. This effect appears above the certain threshold in the optical power Wc circulating in the main mode, when the difference ω0ω1 between frequency ω0 of the main optical mode and frequency ω1 of the idle (Stokes) mode is close to frequency ωm of the mirror mechanical degree of freedom. Coupling between these three modes occurs due to ponderomotive pressure of light in the main mode and Stokes mode and the parametric effect of mechanical oscillation on optical modes. Above the critical value of light power Wc the amplitude of mechanical oscillation is also increasing as the optical power in the idle (Stokes) optical mode gets bigger. However, E. D'Ambrosio and W. Kells have shown [9] that if the anti-Stokes mode (with frequency ω1a=ω0+ωm) is taken into account in the same single dimensional model, then the effect of parametric instability will be substantially lower or even excluded. In [10], [11], [12] an analysis was given based on the model of power and signal recycled LIGO interferometer. It was demonstrated that anti-Stokes mode could not completely suppress the effect of parametric oscillatory instability. As a possible “cure” to avoid the parametric instability it was proposed [13] changing the mirror shape and introducing low noise damping. D. Blair with colleagues proposed an interesting concept of heating test masses in order to vary the curvature radii of mirrors in interferometer and hence to control detuning and decrease the overlapping factor between the optical and acoustic modes [14], [15], [16]. Recently, the instability produced by the optical rigidity was observed in direct experiment [17]. The effect of parametric instability was observed by K. Vahala with collaborators for micro-scale whispering gallery optical resonators [18], [19].

In this Letter we present a detail analysis of parametric instability in signal recycled GEO 600 interferometer and show that in spite of lower optical power in GEO 600 as compared with LIGO, the parametric instability in this interferometer can be observed if detuning Δ=ω0ω1ωm is small. It can be done by changing the frequency of anti-symmetric optical mode (see definition below) of interferometer through varying the position of signal recycling (SR) mirror. It allows using GEO 600 as a testing area to observe precursors of PI and to work out technology to avoid it.

In Section 2 we derive the parametric instability conditions in GEO 600 interferometer. The results obtained are discussed in Section 3. The details of calculations are present in Appendix A.

Section snippets

GEO 600 interferometer

We analyze GEO 600 interferometer with signal recycling (SR) and power recycling (PR) mirrors—see Fig. 1 and notations to it. The interferometer is tuned in resonance and no optical power from the carrier passes through SR mirror. The wave E6 traveling through it is used to detect the signal. Interferometer is pumped through port F5. We make the following simplifying assumptions:

  • Optical losses in all mirrors as well as thermal noises are not taken into account.

  • Transparencies of PR and SR

Discussion and conclusion

Looking at Eqs. (2.32), (2.33) we see that in case of zero detuning the parametric instability may take place in GEO 600 interferometer if factors Q/γmγ± are greater than 1. For parameters from Table 1 we estimate:Qγmγ+1.27×Λ,Qγmγ0.06×Λ,Qprγmγ+2.45×Λ. Hence, one may conclude that chance to observe parametric instability in GEO 600 interferometer is small enough because (a) overlapping factor is usually small (Λ<0.1) and (b) detuning is non-zero in reality and it will also depress parametric

Acknowledgements

We are grateful to Vladimir Braginsky, Bill Kells and David Ottaway for useful discussions on parametric instability problem. Especially we would like to thank Hartmut Grote, as fruitful discussions with him stimulated us to write this Letter and Stefan Hild who provided us with valuable information on GEO 600 parameters. This work was supported by LIGO team from Caltech and in part by NSF and Caltech grant PHY-0353775, by the Russian Agency of Industry and Science: contract No. 5178.2006.2.

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