First Demonstration of Electrostatic Damping of Parametric Instability at Advanced LIGO
Interferometric gravitational wave detectors operate with high optical power in their arms in order to achieve high shot-noise limited strain sensitivity. A significant limitation to increasing the optical power is the phenomenon of three-mode parametric instabilities, in which the laser field in the arm cavities is scattered into higher order optical modes by acoustic modes of the cavity mirrors. The optical modes can further drive the acoustic modes via radiation pressure, potentially producing an exponential buildup. One proposed technique to stabilize parametric instability is active damping of acoustic modes. We report here the first demonstration of damping a parametrically unstable mode using active feedback forces on the cavity mirror. A 15,538 Hz mode that grew exponentially with a time constant of 182 sec was damped using electro-static actuation, with a resulting decay time constant of 23 sec. An average control force of 0.03 nN rms was required to maintain the acoustic mode at its minimum amplitude.
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Introduction Three-mode parametric instability (PI) has been a known issue for advanced laser interferometer gravitational wave detectors since first recognised by Braginsky et al Braginsky et al. (2001), and modelled in increasing detail Zhao et al. (2005a); Strigin and Vyatchanin (2007); Evans et al. (2010); Gras et al. (2010); Vyatchanin and Strigin (2012). The phenomenon was first observed in 2009 in microcavities Tomes and Carmon (2009), then in 2014 in an 80 m cavity Zhao et al. (2015) and soon afterwards during the commissioning of Advanced LIGO Evans et al. (2015). Left uncontrolled PI results in the optical cavity control systems becoming unstable on time scales of tens of minutes to hours Evans et al. (2015).
The first detection of gravitational waves was made by two Advanced LIGO laser interferometer gravitational wave detectors with about 100 kW of circulating power in their arm cavities Abbott and et al (2016). To achieve this power level required suppression of PI through thermal tuning of the higher-order mode eigen-frequency Zhao et al. (2005b) explained later in this paper. This tuning allowed the optical power to be increased in Advanced LIGO from about 5 % to 12 % of the design power, sufficient to attain a strain sensitivity of at 100 Hz.
At the design power it will not be possible to avoid instabilities using thermal tuning alone for two reasons. First the parametric gain scales linearly with optical power and second the acoustic mode density is so high that thermal detuning for one acoustic mode brings other modes into resonance Zhao et al. (2005b); Evans et al. (2015).
Several methods are likely to be useful for controlling PI. Active thermal tuning will minimize the effects of thermal transients Fan et al. (2008); Ramette et al. (2016) and maintain operation near the parametric gain minimum. In the future, acoustic mode dampers attached to the test masses Gras et al. (2015) could damp acoustic modes. Active damping Miller et al. (2011) of acoustic modes can also suppress instabilities, by applying feedback forces to the test masses.
In this letter we report on the control of a PI by actively damping a 15.54 kHz acoustic mode of an Advanced LIGO test mass using electro-static force actuators. First we review the physics of PI and the status of PI control in LIGO. Then we discuss the electrostatic drive system at LIGO and how it interacts with the test mass modes. Then we summarise the experimental configuration, report successful damping observations, and discuss the implications for high power operation of Advanced LIGO.
Parametric Instability The parametric gain , as derived by Evans et al Evans et al. (2010) is given by;
Here is the quality factor (Q) of the mechanical mode , is the power in the fundamental optical mode of the cavity, is the mass of the test mass, is the speed of light, is the wavelength of light, is the mechanical mode angular frequency, is the transfer function for an optical field leaving the test mass surface to the field incident on that same surface and is the spatial overlap between the optical beat note pressure distribution and the mechanical mode surface deformation.
It is instructive to consider the simplified case of a single cavity and a single optical mode to understand the phenomena. For a simulation analysis including arms and recycling cavities see Gras et al. (2010); Evans et al. (2010) and for an explanation of dynamic effects that may make high parametric gains from the recycling cavities less likely see Zhao et al. (2015). In the simplified case we consider the mode as it dominates the optical interaction with the acoustic mode investigated here;
Here is the half-width at half maximum of the optical mode frequency distribution, L is the length of the cavity, is the spacing in frequency between the mechanical mode and the beat note of the fundamental and TEM optical modes. In general the parametric gain changes the time constant of the mechanical mode as in Equation 3. If the parametric gain exceeds unity the mode becomes unstable.
Where is the natural time constant of the mechanical mode and is the time constant of the mode influenced by the opto-mechanical interaction. Thermal tuning was used to control PI in Advanced LIGO’s Observation run 1 and was integral to this experiment, so will be examined in some detail. Thermal tuning is achieved using radiative ring heaters that surround the barrel of each test mass without physical contact as in Figure 2. Applying power to the ring heater decreases the radius of curvature (RoC) of the mirrors. This changes the cavity g-factor and tunes the mode spacing between the fundamental () and higher order transverse electromagnetic () modes in the cavity, thereby tuning the parametric gain by changing in Equation 2.
Figure 1 shows the optical gain curve (Equation 2) for the mode, with the ring heater tuning used during Advanced LIGOâs first observing run Abbott et al. (2016). With no thermal tuning, the optical gain curve in Figure 1 moves to higher frequency, decreasing the frequency spacing with mode group E. This leads to the instability of this group of modes. (Note that the mirror acoustic mode frequencies are only weakly tuned by heater power, due to the small value of the fused silica temperature dependence of Young’s modulus). If the ring heater power is increased inducing approximately 5 m change in radius of curvature, the beat note gain curve in Figure 1 moves left about 400 Hz, decreasing the value for mode group A, resulting in their instability. The mode groups C and D are stable as the second and fourth order optical modes that might be excited from these modes are far from resonance. Mode Group B is also stable at the circulating optical power used in this experiment presumably due to either lower quality factor or lower optical gain of the TEM mode as investigated in Barriga et al. (2007). If the power in the interferometer is increased by a factor of 3 there will no longer be a stable region. Mode group A at 15.00 kHz and group E at 15.54 kHz will be unstable simultaneously.
Electrostatic Control Electrostatic control of PI was proposed Ju et al. (2009) and studied in the context of the LIGO electrostatic control combs by Miller et al Miller et al. (2011). Here we report studies of electrostatic feedback damping for the group E modes at 15.54 kHz.
The main purpose of the electrostatic drive (ESD) is to provide longitudinal actuation on the test masses for lock acquisition Miyakawa et al. (2006) and holding the arm cavities on resonance. It creates a force between the test masses and their counterpart reaction masses, through the interaction of the fused silica test masses with the electric fields generated by a comb of gold conductors that are deposited on the reaction mass. The physical locations of these components are depicted in Figure 2. Detail of the gold comb is shown in Figure 3 along with the force density on the test mass.
The force applied to the test mass is dominated by the dipole attraction of the test mass dielectric to the electric field between the electrodes of the gold comb. Some portion of this force that couples to the acoustic mode as;
Here is the force coefficient for a single quadrant, while and are the voltages of the ESD electrodes defined in Figure 3. The overlap between the ESD force distribution and the displacement of the surface for a particular acoustic mode can be approximated as a surface integral derived by Miller Miller et al. (2011):
If a feedback system is created that senses the mode amplitude and provides a viscous damping force using the ESD, the resulting time constant of the mode is given by;
Here is the gain applied between the velocity measurement and the ESD actuation force on a mode with time constant and effective mass . Reducing the effective time constant lowers the effective parametric gain.
The force required to reduce a parametric gain to an effective parametric gain when the mode amplitude is the thermally excited amplitude was used by Miller Miller et al. (2011) to predict the forces required from the ESD for damping PI,
at the thermally excited amplitude , where is the Boltzmann constant and temperature.
Feedback Loop Figure 4 shows the damping feedback loop implemented on the end test mass of the Y-arm (ETMY). The error signal used for mode damping is constructed from a quadrant photodiode (QPD) that receives light transmitted by ETMY. By suitably combining QPD elements, we measure the beat signal between the cavity mode and the mode that is being excited by the 15,538 Hz ETMY acoustic mode. This signal is band-pass filtered at 15,538 Hz, then phase shifted to produce a control signal that is 90 degrees out of phase with the mode amplitude (velocity damping). The damping force is applied, with adjustable gain, to two quadrants of the ETMY electro-static actuator.
Results PI stabilization via active damping was demonstrated by first causing the ETMY 15,538 Hz to become parametrically unstable; this was done by turning off the ring heater tuning, so that the mode optical gain curve better overlapped this acoustic mode, as shown in Figure 1. When the mode became significantly elevated in the QPD signal, the damping loop was closed with a control gain to achieve a clear damping of the mode amplitude and a control phase optimised to degrees of viscous damping. The mode amplitude was monitored using the photodetector at the main output of the interferometer (labelled OMC-PD in Figure 4), as it was found to provide a higher signal-to-noise ratio than the QPD.
The results are shown in Figure 5, which plots the mode amplitude during the unstable ring-up phase, followed by the ring-down when the damping loop is engaged. From the ring-up phase, we estimate the parametric gain to be from Equation 3. With the damping applied,
the effective parametric gain is reduced to a stable value of . The uncertainty is primarily due to the uncertainty in the estimate of which was obtained by the method described in Evans et al. (2015).
At the onset of active damping (time t = 0 in Figure 5), the feedback control signal produces an estimated force of 0.62 nN rms (at 15,538 Hz). As the mode amplitude decreased the control force dropped to a steady state value of 0.03 nN rms. Over a 20 minute period in this damped state, the peak control force was 0.11 nN peak.
|Q factor of 15,538 Hz mode|
|P||100 kW||Power contained in arm cavity|
|15,538 Hz||Frequency of unstable mode|
|M||40kg||mass of test mass|
|0.17||effective mass scaled ESD overlap factor for 15,538 Hz mode|
|1064 nm||laser wavelength|
|N/V^2ESD quadrant force coefficient|
|L||4km||Arm cavity length|
|400V||Bias voltage on ESD|
|[-20,20]V||ESD control voltage range|
Discussion The force required to damp the 15,538 Hz mode when advanced LIGO reaches design power can be determined from the ESD force used to achieve the observed parametric gain suppression presented here, combined with the expected parametric gain when operated at high power.
The maximum parametric gain of the 15,538 Hz mode (where ) at the power level of these experiments is estimated given an estimated de-tuning of with zero ring heater power. At full design power the maximum gain will be . To obtain a quantitative result, we set a requirement for damping such that the effective parametric gain of unstable acoustic modes after damping be .
Using Equation 10, the measurements of and , the maximum force required to maintain the damped state at high power is 1.5 nN rms. Prior to this investigation Miller predicted Miller et al. (2011) that a control force of approximately would be required to maintain this mode at the thermally excited level.
The PI control system must cope with elevated mode amplitudes as the PI mode may build up before PI control can be engaged. There is therefore a requirement for some control range or safety factor such that the control system will not saturate if the mode amplitude is a multiple of the safety factor times the damped state amplitude. The average ESD drive voltage over the duration the mode was in the damped state was 0.42 mV rms, however during this time it peaked at 1.4 mV peak out of a control range, leading to a safety factor of more than 10,000. At high power the safety factor will be reduced by the required force ratio of Equation 10 resulting in an expected safety factor of 310.
As the laser power is increased, other modes are likely to become unstable. The parametric gain of these modes should be less than the gain of mode group E provided the optical beat note frequency used in these experiments is maintained. However these modes may also have lower spatial overlap with the ESD. Miller’s simulation Miller et al. (2011) show some modes in the 30-90 kHz range will require up to 30 times the control force required to damp the group E modes. Even in this situation the PI safety factor is approximately 10.
Conclusion We have shown for the first time electrostatic control of parametric instability. An unstable acoustic mode at 15,538 Hz with a parametric gain of was successfully damped to a gain of , using electrostatic control forces. The damping force required to keep the mode in the damped state was 0.03 nN rms. The prediction through FEM simulation was that the ESD would need to apply approximately six times this control force to maintain the mode amplitude at the thermally excited level. At high power it is estimated that damping the 15.54 kHz mode group to an effective parametric gain of 0.1 will result in a safety factor 310. It is predicted that unstable modes that are most problematic to damp will still have a safety factor of 10.
Acknowledgments The authors would like to acknowledge the entire LIGO Scientific Collaboration for the wide ranging expertise that has contributed to these investigations. LIGO was constructed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation, and operates under Cooperative Agreement No. PHY-0757058. Advanced LIGO was built under Grant No. PHY-0823459. This paper has LIGO Document Number LIGO-P1600090. The corresponding author was supported by the Australian Research Council and the LSC fellows program.
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