Radiation pressure induced dynamics in a suspended Fabry-Perot cavity

1. Radiation pressure induced dynamics in a suspended Fabry-Perot cavity Thomas Corbitt, David Ottaway, Edith Innerhofer, Jason Pelc, and Nergis Mavalvala

2. Feedback model of optical rigidity P force + x f POR f x k

3. Feedback model Modified response is identical to a harmonic oscillator with a modified frequency and damping constant, under some (not so good) assumptions

4. PD Experiment Vacuum 3.6 W 2 kW PSL EOM QWP 250 gram Frequency control (high freq.) Length control (low freq.) PD X 25.2 MHz PDH / T Length VCO

5. Optical spring resonance • For bulk motion of the mirrors, the dominant mechanical restoring force is gravitational force from the suspensions, with frequency ~1 Hz. • Predicted optical rigidity should give optical spring resonance ~ 80 Hz, so the gravitational restoring force is negligible • We looked for the resonance, but...

6. Looking for the optical spring • Injected highest available power level into locked cavity • Detuned to where the maximum optical rigidity was expected • Looked for the optical spring • After running for a short time (<1 min), observed large oscillations in the error signal at 28 kHz • Already knew this was the drumhead mode frequency • Fluctuations disappeared when we went back to the center of the resonance

7. Nuisance headache • Tested on both sides of the resonance • The mode only became excited on one side • Tested at various power levels • Either the mode became excited until the fluctuations were ~ linewidth large • Or it did not become excited at all below some power level • Tested with different gain in the frequency feedback path • Found that we could (de-)stabilize the mode by playing with this • The mode remained unstable when the frequency path had no gain  instability not a feedback effect

8. Parametric Instability!!! • Instability depends on power and detuning • Is not a feedback effect • Must be a parametric instability • The drumhead motion of the mirror creates a phase shift on the light • The phase shift is converted into intensity fluctuations by the detuned cavity, which in turn push back against the drumhead mode • Arises from the same optical rigidity, just applied to a different mode • For this mode, the optical rigidity is much weaker than the mechanical restoring force, so how can it destabilize the system?

9. Parametric Instability Model

10. Implications for Advanced LIGO • For the parametric instability observed here • The mechanical mode frequency (28 kHz) is within the linewidth of the cavity (75 kHz) • This is different from the type of instability that people worry about with Advanced LIGO, e.g. • Occurs when the mechanical mode frequency is outside the linewidth of the cavity • Higher order spatial modes of the cavity must overlap in frequency space with the frequency of the mechanical mode

11. Measuring the Parametric Instability • Measure the PI as a function of power and detuning • For regions where the mode is unstable, measure the ring-up time (few seconds). • For regions where the mode is stable, first go to an unstable region, ring-up, then rapidly go to stable region and measure ring-down time. • Do the measurements with 0 gain in the feedback paths at 28 kHz to prevent any interference • Frequency feedback path turned off • Length control had a 60 dB notch filter at 28 kHz (UGF at ~1 kHz). • Measurements show • R scales linearly with power. • R shows reasonable agreement with predictions for dependence on detuning

12. Parametric Instability Results

13. Damping the PI VCO gain turned up

14. Back to the Optical Spring • To have a large optical spring frequency, we wanted to use full power • Locked the frequency path with ~50 kHz bandwidth to have sufficient gain at 28 kHz to stabilize the unstable mode at 28 kHz • Now that the parametric instability was identified and damped, we returned to the optical spring • The resonance was expected at ~80 Hz, well within our servo bandwidth, so • Inject signal into feedback paths • Measure transfer function from force (either length/frequency path) to error signal (displacement) to measure the modified pendulum response

15. Optical Spring Measured • Phase increases by 180˚, so resonance is unstable! • But there is lots of gain at this frequency, so it doesn't destabilize the system

16. Next phase • Use both input (250 g) and end (1 g) mirrors of the ponderomotive squeezing experiment in a suspended 1 m long cavity • R < 50 at full power • <1 MW/cm2 power density • Optical spring resonance at > 1 kHz • Final suspension for 1 gm mirror not ready yet, so • Glue mirror to small optic blank • Double suspension? • Goals for next stage • Get optical spring out of the servo bandwidth • See instability directly and damp it • See noise reduction effects • Build up to full interferometer for ponderomotive squeezing