## page was renamed from Upgrade 09/GreenLock/Noise Requirement
 . {{attachment:greenskeleton.jpg}}       



== Noise Requirement ==
I'd say the arm cavity length should not change more than 1/100 of the resonance width in the time scale of 10sec.<<BR>> For this to be fulfilled, the RMS fluctuation of the cavity length must be suppressed below 1e-11m level.<<BR>> ''Does this make sense ?''

=== Noise Sources ===
==== PLL phase noise ====
==== Phase noise from the cavity common mode motion ====
When the cavity is locked to the green laser, the differential motion of the two mirrors will be suppressed by the servo.<<BR>> However, the common motion of the cavity mirrors will not be suppressed. This common motion will show up as phase noises <<BR>> of the lasers. The time derivative of phase noise is equivalent to frequency noise. The equivalent displacement noise seen <<BR>> by the cavity is dL=(w*x*L)/c, where w is the angular frequency, x is the displacement noise spectrum of the common motion, <<BR>> L is the length of the cavity and c is the speed of light (see [[attachment:PhaseNoise.pdf|attachment:PhaseNoise.pdf]] for derivation).

{{attachment:Cavity-Common-Diff.png}}

The two lasers (PSL and green) see the same motion of the cavity but from the opposite sides. Hence, the effect of this phase <<BR>> noise to the two error signals of the green and the PSL lasers will be 180 deg. out of phase. The feedback from the green laser to <<BR>> the cavity length will, therefore, create a noise for the PSL laser.

In order to estimate the contribution of this noise to the green lock, I plotted the estimated phase noise in the following figure.<<BR>>  {{attachment:PhaseNoise.png}}

I first took a spectrum of ETMX OSEM pos signal to see the motion of the mirror with damping.<<BR>> The blue curve in the figure shows the calibrated OSEM spectrum using the well known 2V/mm OSEM <<BR>> calibration and the whitening filter shape (3Hz zero, 30Hz and 100Hz pole).<<BR>> However, OSEM signal is not a good measure of the seismic noise below the pendulum resonant frequency<<BR>> because the suspension cage and the mirror move together at low frequencies.<<BR>> As a tentative solution, I put a filter to make the spectrum look like 1/f^2 below 0.8Hz.<<BR>> This is diffinitely a hacky solution, and should be replaced with a correctly measured <<BR>> seismic spectrum.

The estimated seismic motion was converted to the phase noise using the above formula.<<BR>> The RMS displacement noise above 0.1Hz is about 3e-12 m, which satisfies the requirement for <<BR>> the green lock stability (1e-11m).