Observation of Parametric Instability in Advanced LIGO

Parametric instabilities have long been studied as a potentially limiting effect in high-power interferometric gravitational wave detectors. Until now, however, these instabilities have never been observed in a kilometer-scale interferometer. In this Letter, we describe the first observation of parametric instability in a gravitational wave detector, and the means by which it has been removed as a barrier to progress.

Figure 1

Parametric instabilities transfer energy from the arm cavity field to a mechanical mode of an interferometer test mass. Since the rate of energy transfer depends on the amplitude of the excited mechanical mode, this can be a runaway positive feedback process.

Figure 2

The excitation due to parametric instability was first observed as an exponentially growing feature in the detector’s primary output (the gravitational wave channel). The feature is aliased into the detector band by the 16384 Hz sample rate of the digital system which causes it to appear at 845 Hz. The growing instability eventually causes saturation of the electronic readout chain, which appears as broadband contamination of the detector output channel (visible between 2500 and 3000 s). In the above data, the power into the interferometer was decreased before saturation caused the interferometer control systems to fail (see Fig. 4).

Figure 3

The test mass mechanical mode and cavity optical mode responsible for the observed parametric instability are shown. The left panel is the test mass front surface displacement due to the mechanical mode, while the right panel is the radiation pressure induced displacement (red is positive and blue negative in both panels). Both of these modes occur at 15.5 kHz and they have an overlap factor of $B_{m,n}=0.1$.

Figure 4

The amplitude of the excitation shown in Fig. 2 is fitted at times with different values of $P_{\text{arm}}$ to find its growth (or decay) time scale as a function of power. The growth of the PI is clearly visible above the noise after 1000 s, with an $e$-folding growth time of 240 s, until 2000 s at which point the readout electronics begin to saturate. A little more than 4000 s into the plotted data, the excited mechanical mode is seen decaying with $\tau =-900\mathrm{s}$. According to Eq. (2), these data imply a threshold power of 25 kW and $Q=12\times {10}^6$ for this mode.

Figure 5

The maximum gain expected at full power, $P_{\text{arm}}=800\mathrm{kW}$, for all potentially unstable modes (90% confidence value for a single test mass). Since the mechanical mode frequencies and optics parameters are known with limited precision, the parametric gain of a particular mode cannot be computed exactly, and Monte Carlo methods must be used [31]. The “90% confidence maximum” shown here is the value for which 90% of the 342 881 Monte Carlo computations gave a lower gain. Notably, the observed unstable mode is the mode with the highest predicted parametric gain (marked with a green star), and the observed gain is about a factor of 3 higher than the value shown here ($R=2$ at 50 kW, or $R=32$ at 800 kW). This is not so much bad luck as a trials factor; there are four test masses and many potentially unstable modes, and the statistics shown in Fig. 7 confirm quantitatively that the observation of at least one unstable mode was to be expected.

Figure 6

The number of unstable modes as a function of change in arm cavity mirror radius of curvature (ROC), as indicated by the color scale to the right of each plot. With 100 kW of circulating power, as expected for the first observing run, small changes in mirror ROC can be used to avoid parametric instability. Higher power levels, however, increase parametric gain; note the different color scales used on the left and right panels. At Advanced LIGO’s design operating power of $P_{\text{arm}}=800\mathrm{kW}$, ROC adjustments can at best be used to reduce the number of unstable modes.

Figure 7

The probabilities of having various numbers of unstable modes anywhere in the interferometer (with four test masses), as a function of circulating power, $P_{\text{arm}}$. Each curve gives the probability that at least $N=\{1,2,3,5,\dots \}$ unstable modes are observed in an Advanced LIGO interferometer at a given level of circulating power in the arm cavities. The vertical gray lines mark the threshold power of the mode first observed to be unstable, near 25 kW of circulating power, the power level at which the first observing run of Advanced LIGO is planned, roughly 100 kW, and the design power level of 800 kW. These data indicate that more than 40 modes will likely need to be damped or otherwise defused in order to operate at 800 kW.