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

Feedback model of optical rigidity force + f x POR f x k

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

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

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...

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

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

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?

Parametric Instability Model

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

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

Parametric Instability Results

Damping the PI VCO gain turned up

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

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

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