I think the precision due to the loop gain uncertainty is something like 0.1% at 0.1 Hz. It's not the issue.
The real issue was the loud motion of MICH, which degrades the coherence of the measurement.
Also last night I tried the fringe hopping technique and gave it up for several reasons.
(uncertainty due to the loop gain)
When MICH is locked, the signal at C1:LSC-MICH_OUT can be expressed in frequency domain by
MICH_OUT = G / (1+G) * (1 / A) * X + G / (1+G) * (1 / H) * (1 / A) * S, [1]
where G is the open loop gain, A is the actuator response, H is the sensor transfer function (constant factor),
X is the natural (unsuppressed) motion of MICH and S is an excitation injected at C1:LSC-MICH_EXC.
When the natural motion of MICH X is smaller than the excited displacement S/H, dividing MICH_OUT by S gives
[Transfer function] = S / MICH_OUT
= (1+G) /G * H * A
At low frequency the open loop gain is always big, so that the transfer function can be approximated to
[Transfer function] ~ H *A
This approximation is valid with a precision of 1/G.
In my case yesterday, the open loop gain at 0.1Hz was about 103 or more than that, so the uncertainty due to the loop gain was 0.1% or even less.
(Effect from the MICH motion)
In the equation [1], it is shown that the MICH motion X shows up together with the excitation signal.
Actually this MICH motion term was not completely negligible and eventually this term disturbs the measurement resulting in a low coherence.
In order to get a high coherence in the measurement, X should be smaller than the excited displacement S/H,
X << S / H
This the reason why I had to inject a big excitation signal. Although the coherence around 1Hz turned out to be still low due to the loud natural motion in MICH.
The excitation was already close to 0.1 um level in terms of peak-to-peak displacement, and I wasn't able to increase it any more because the MICH signal would run into a nonlinear regime.
In the worst case I lost the lock due to a too much excitation.
(Fringe hopping technique)
Actually I tried and gave up this technique. That's why I did the in-loop measurement.
My feeling is that this technique is not suitable for the 40m.
What I tried was to flip the sign of the MICH control such that the fringe hops from the dark fringe to the neighbor bright fringe or vice versa.
Difference in the control signal (C1:LSC-MICH_OUT) was supposed to give us the amount of signal which drives the actuator by exactly quarter of the laser wave length.
However this technique turned out to be not good because
(1) BS actuator is too strong
=> expected difference in the control signal is quite small.
=> \lambda / 4 / A ~ 12 counts, where A is the actuator DC response of about 2.2e-8 [m/counts].
(2) MICH motion was too loud
=> I saw such a tiny 12 counts difference in the control signal, but once the hopping is done the control signal immediately fluctuated and it was really hard to precisely measure it.
=> It's simply because MICH was loud, and the actuator tried to suppress the motion and it resulted such an immediate signal fluctuation in the control signal
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This seems like an error prone method for DC responses due to the loop gain uncertainty. Better may be to use the fringe hopping method (c.f. Luca Matone) or the fringe counting method
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