Summary
In 2arms + MICH configuration, residual motion of the cavity will couple with MICH signal. When cavity length change, the reflectivity of cavity also change. And that cause the phase shift in reflected light. That phase shift is detected in MICH signal. When we try to lock the DRMI + arm, that coupling will be problem for lock acquisition. For practice to estimate that coupling, I estimated the coupling between the cavity motion and the AS55Q signal.

What I did

- Measurement steps
  I did the same measurement as that of this entry. For the estimation below steps are needed. The detail of each step will be written below.
  --Measurement and calibration of the AS55Q error signal with MICH + 2arms locked by ALS control
  --Measurement of the ALS in-loop noise and estimation of residual motion of the cavities.
  --Calibration of the coupling from residual arm motion to AS55Q signal

- Calibration of  the AS55Q signal
1. Sensor gain estimation
  We used the same method as the previous entry,
  We excited the BS at 580 Hz with a given amplitude (Vin). We enabled the notch filter at 580 Hz in the LSC MICH servo. We measured  the peak height (Verr) of the AS55Q error signal. We used the actuator response (A_bs) of BS measured in this entry.
  We can get the sensor gain (H) of AS55Q in unit of count/m

          Verr    1
   H = ------- -------
          Vin   A_bs

By this calculation H = 4.2e+07.

2. Fitting of OLTF for the MICH loop
  We measured the OLTF of the MICH loop. Modelled OLTF is fitted into the measurement data. That modelled OLTF includes the actuator response of BS, the MICH servo filters, DAI,DAA,AI,AA filters, the TF of sample and hold circuit. (About DAI, DAA filters and S/H circuit please read this entry. About AI,AA filters please read this entry)  Also I put time-delay into that OLTF. I estimated that time-delay and the gain of OLTF by fitting.  The time delay was 311usec.

3. Estimation of the MICH free running noise
 With modeled OLTF, I estimated the MICH free running noise.

Estimation of the coupling from residual cavity motion to AS55Q signal
 The ALS in-loop noise data has the unit of Hz/rHz (disturbance of the cavity resonant frequency). By multiplying L_arm/f_laser we can convert the unit to m/rHz (disturbance of the cavity length) .
 I used the same coupling constant between residual motion of cavity and MICH noise as this entry. For estimation of the coupling constant, we excited ETMs  and measured the TF from excitation signal to AS55Q error signal.  I assumed the cavity pole as 4000 Hz. The result is discussed below

Discussion

  ALS in-loop noise include the sensor noise. in high frequency region the in-loop noise is dominated by the sensor noise. So in this region in-loop noise does not mean actual residual motion of the cavity.  And this sensor noise pushes the mirror. So we have to estimate the actual motion of the cavity by multiplying the servo transfer function of the control in this region.

 I made 2 plots. Both include the MICH free running noise and estimated coupling noise from both arms. In one plot, for estimation of the coupling I multiplied only coupling constant to calibrated in-loop noise of the ALS loop. In another plot,  I multiplied coupling constant and OLTF of ALS loop in order to estimate the actual motion of the cavity.  If the 3 curves are coincide in first plot, that means the ALS in-loop noise is same as the residual cavity motion in that region and the MICH free running noise is dominated by coupling from residual cavity motion. If those curves are coincide in second plot, that means the ALS in-loop noise is sensor noise in that region.

 Above 40 Hz, the 3 curves are totally in coincident in first plot. On the other hand in second plot the 3 curves look similar in this region. That may mean above 40 Hz the ALS noise are dominated by sensor noise and MICH free running noise is dominated by the coupling from residual cavity motion.  Also in the region between 10 Hz and 40 Hz, the MICH free running noise seems to be dominated by coupling from cavity motion.

Figure 1

Figure 2

In second plot, the coupling from cavity motion is overestimated. It's possibly because of overestimation of coupling constant, but I'm not sure.
Koji mentioned that we should measure the residual motion of the cavity by using POX and POY. Now the ALS is much more stable than before, so I think we can easily do the measurement again with out of loop measurement. That will be more strait forward measurement.