It seems that we could enhance our A2L decoupling by including the quadratic terms such as DH_P(t) * CS_P(t) when doing offline noise subtraction.
Please see the attached plot for an example.
Details:
Note that a2l noise in t-domain is given by
dL(t) = dy(t) * dTheta(t),
where dy is the spot position and dTheta is the angular motion of a mirror. In the online A2L feedforward (as done in the SUS L2 stage), we do
dL_ff(t) = dy(DC) * dTheta(t),
i.e., subtracting the dominating 'linear' coupling. The dither locking basically does the same thing as it servos dy(DC) to a fixed value.
On the other hand, we also have
dy(t) ~ dy/dTheta * dTheta( t - t_delay),
where dy/dTheta is a constant determined completely by the cavity's geometry. For the off-resonance 01/10 modes, the delay time is on the order
t_delay ~ 2L/c ~ 3x10^{-5} sec,
which is effectively 0 for the A2L coupling we are interested in.
As a result, it suggests that the A2L noise will contain terms like
dL(t) ~ dTheta(t)**2,
in the time domain.
The argument above suggests that in the offline data cleaning, in addition to include the linear ASC terms, such as DH_P(t), we can also include quadratic terms that are the products of ASC outputs in the time-domain, e.g., DH_P(t) * CS_P(t).
In the attached plot, the blue trace is the calibrated DARM ('GDS-CALIB_STRAIN') at gps: 1236207018 + [0, 2048] sec. I took 8 sec of data per fft and 50% overlap, thus 511 averages in total.
In the orange trace is the DARM after removing all the linear ASC coupling using a freq-domain MISO-coherence-based subtraction. We don't see any significant improvement over the noise, which is as expected as the soft dithering loops should zero the linear coupling.
In the green trace, we further include quadratic ASC outputs and again do the freq-domain MISO-coherence analysis. This seems to improve the subtraction in the 10-30 Hz band where the ASC noise dominates.
The caveat is that the subtraction done so far are all in the freq-domain. I haven't tested how-well a t-domain Wiener filter could reproduce the result. More importantly, due to the amount of clean lock stretches available, we haven't checked how stationary the quadratic coupling is from one time to another.
Nevertheless, the initial result seems promising, and including those quadratic terms should not be technically hard in the offline subtraction.
Gabriele, Hang
Gabriele pointed out that the f-domain subtraction using coherence could easily lead to over subtraction, as the finite amount of average could over-estimate the coherence. For example, the improvement above 30 Hz in the previous attached plot could hardly be real as the ASC noise should have been rolled off there.
We thus looked at a longer data stretch of 10240 sec (~ 3 hrs; 5 times what we used in the previous study) starting at gps 1236511818. Again we use 8 sec per fft and 50% overlaps, and therefore 2559 averages. This should mitigate the errors due to finite averaging. The result is attached to this entry.
In the first plot we show the DARM noise spectra before and after subtracting the linear/quadratic ASC channels.
In the second plot we present the largest contributions' projection to DARM (ranked by the contribution at 25 Hz).
In the last one we compute the SISO coherence between [CH_Y(t)xCS_P(t)] and DARM as a sanity check for result plotted in the second figure.
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Some conclusions based on the plot:
1. We still see a significant (visible by eyes) improvement in the 15-30 Hz band by including the quadratic terms such as DH_P(t) x CS_P(t). The level of noise reduction is roughly comparable to what we had in the previous entry in this band.
2. Below 20 Hz the dominating nonlinear correction comes from [CH_P(t) x DH_P(t)] which is more or less expected. On the other hand, the largest contribution at 25 Hz is from [CH_Y(t) x CS_P(t)]. This is a bit surprising. Nonetheless, we also looked at the SISO coherence and saw a similar amount of projection to DARM. Thus the coupling should be real. It suggests that we may have large P/Y and C/D cross-coupling simultaneously.
3. Above 30 Hz, the quadratic subtraction matches the linear subtraction. This should be viewed as a reduction in the systematic error in our approach.
4. In the 10-15 Hz band, even linear subtraction can still improve the noise performance. This is an indication that in addition to the geometrical a2l coupling (that is freq independent), we also have a path of angle->power->length coupling (that scales ~ 1/f^2 and thus is visible only at low frequencies; this path also cannot be subtracted by the dithering soft loop which handles only the geometrical one). This is consistent with the previous observations that as we decreased the dithering frequency (thus locking pt was more biased by the radiation P effect), the ASC->DARM coupling got worse at ~20 Hz.