Written by Yuta

Xgreen frequency noise spectra from alignment fluctuation (see alog #9429, #9384 for more information) was calculated using oplev spectra of ITMX.
Even if we keep our sideband frequency to be 24.407079MHz and set the demodulation phase to maximize 00 PDH slope, the frequency noise is ~40 Hz in RMS when DC misalignment is 1 urad. Maybe we can work it out without using sideband frequency / demodulation phase tuning technique.

[Method]
1. Get calibrated ITMX oplev spectra (Thanks to Jeff !).

2. Frequency noise from misalignment(HOM) can be written as

fn(t) = k (a(t) + a0)^2

where fn(t) is the frequency noise, k is the conversion factor, a(t) is alignment fluctuation and a0 is DC misalignment. Note that this effect is quadratic and estimated k=1e15 Hz/rad^2 if we keep our setting as is (see alog #9429) and we assume differential angular motion of ITM and ETM.
The spectrum of fn(t) can be calculated using the spectra of a(t) from the following formula.

fn(f) = k^2 * (conv(a(f),a(f)) + 2*conv(a(f),fliplr(a(f)))) + 2*k*a0*a(f)

where conv is convolution and fliplr gives flipped array. conv(a(f),a(f)) gives upconversion term and 2*conv(a(f),fliplr(a(f))) gives down conversion term.
(see this document for derivation if you can read Japanese)

[Result]
1. ITMXOplevspectra.png: Measured ITMX angular motion from oplev. Note that spectra above 0.5 Hz is not measuring the actual motion.

2. HOMfreqnoise_0urad.png: Frequency noise from HOM in the case when DC alignment is perfect. This gives 0.3 Hz RMS.

3. HOMfreqnoise_1e-1urad.png: Frequency noise from HOM in the case when DC alignment is off by 0.1 urad. This gives ~4 Hz RMS.

4. HOMfreqnoise_1e-1urad.png: Frequency noise from HOM in the case when DC alignment is off by 1 urad. This gives ~40 Hz RMS.