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
force + f x
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. 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

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