1. Measurement of radiation pressure induced dynamics
Thomas Corbitt, David Ottaway, Edith Innerhofer, Jason Pelc, Daniel Sigg, and Nergis Mavalvala
Based on LIGO-P R or gr-qc/ and some new data

2. Why radiation pressure?
Optical systems that are radiation pressure dominated
Study modified mirror oscillator dynamics
Manipulate optical field quadratures
Phase I – cavity with two 250 g suspended mirrors, finesse of 1000, ~4 W of input power
Phase II – cavity with one 250 g and one 1 g suspended mirror, finesse of 8000, ~1 W of input power
Ultimate goal – quantum-limited radiation pressure for ponderomotive squeezing interferometer

3. Unified Feedback Model
Mechanical oscillator
Optical feedback
Optical spring
Parametric Instability
Electronic servo
Force (F)
Position (x) F x

4. A Simulink Model

5. Properties of optical springs
Optical rigidity
Modified dynamics
Noise suppression

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

7. 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

8. Phase II Cavity
Use 250 g input and 1 g end mirror in a suspended 1 m long cavity with goal of
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
Double suspension
Goals for this stage
See noise reduction effects
Get optical spring out of the servo bandwidth
See instability directly and damp it

9. Double suspension for mini mirror (the “MOS”)

10. Noise Suppression

12. Optical spring at 530 Hz Data (red) vs. model (blue)

13. And last night... 2 kHz!
2 kW circulating

14. The End

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

16. While 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

17. 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?

18. Parametric Instability Model
For drumhead mode, optical restoring force much smaller than mechanical restoring force.

19. 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

20. Parametric Instability Results

21. Damping the PI
VCO gain turned up

22. 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

23. 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...

24. 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