J. Kissel (with input from S. Dwyer, J. Driggers, and C. Cahillane)

I gathered a set of transfer functions between the HAM1 Table Top "TT" L4Cs and the beam pitch DOF of the "DC" "QPD" REFL WFS while I drove HAM1 HPI in Z (vertical). This is in prep for the upcoming HAM1 vent where we intent (among other things) mute the blade springs on the RMs, so we can use this measurement as the metric for success in killing the 1 - 10 Hz Vertical to Pitch dynamics of the HTTS that we've found.

The good news (though, really it's bad news) is that our measurement looks virtually identical to LLO's "RM blades unlocked" trace in LLO aLOG 59677. We've confirmed it elsewhere, but this is more damning evidence that the physical mechanism for coupling is the same at both sites, and gives us confidence that taking the same action will yield the same result (however dissatisfying that we don't understand it physically).

Method
To get light on the REFL WFS, 
   - the IFO is "freshly unlocked," 
   - the PSL power input at 2W, and 
   - the IMC locked. 
   - PR3 is misaligned such that the IFO does not interfere with the measurement. 
   - The DC1 and DC2 ASC centering loops were ON (by just using the same configuration as normal, all we had to do was turn the inputs ON). 
This is as best I could understand/replicate of the measurement taken.

The amount of excitation I drove above ambient noise in order to get good coherence -- namely, a 500 [ct] amplitude uniform gaussian noise excitation in H1:HPI-HAM1_ISO_Z_EXC, with a 5th order elliptic bandpassing 1 to 50 Hz with -60 dB suppression outside the band. 

We also drove the same excitation while the full IFO was running, and it did not break the lock. We have that data in the can for future analysis, but I figure this aLOG is already information overload, and the full-IFO situation will change after we mute the RMs, so no point in scrutinizing the data right now.

The templates live in 
    /ligo/home/jeffrey.kissel/2022-06-06/
        2022-06-06_H1HPIHAM1_to_ASC_TF_HPIEXCON_OnlyIMC_PRMAligned_PR3MisAligned.xml   # this one made the plots attached to this aLOG, and should be what we re-use in-air
        2022-06-06_H1HPIHAM1_to_ASC_TF_HPIEXCOFF_OnlyIMC_PRMAligned_PR3MisAligned.xml  # this one I used to make statements about the ambient situation

        2022-06-06_H1HPIHAM1_to_ASC_TF_HPIEXCOFF_FullLock.xml
        2022-06-06_H1HPIHAM1_to_ASC_TF_HPIEXCON_FullLock.xml

RESULTS
Expanding on what Anamaria, Jorge, Joe Hanson, Valera and show, I've made an effort to calibrate the sensors in the measurement, which you can see in the DETAILS below. In doing so, I can show:
    - 2022-06-06_H1HPIHAM1_to_ASC_TF_HPIEXC.png: The error signal (and therefore the excitation channel) should be in units of displacement [nm], but I wasn't *super* sure, so I left it in [cts], but its counts of those channels.
    - 2022-06-06_H1HPIHAM1_to_ASC_TF_TTL4Cs_DuringHPIEXC.png: Here, a properly calibrated amplitude spectral density of the displacement of the table in Z, according to the table-top L4Cs. NOTE: It looks very fishy that 
        (a) from 0.1 to 1 Hz, the ambient table motion is higher than the L4C noise, and
        (b) from 0.1 to 1 Hz, this "aggressively band-passed below 1 Hz" excitation creates a factor of 5 more motion, BUT
      I'm mollified by the fact that the high frequency asymptote matches our standard model of the L4C noise.
    - 2022-06-06_H1HPIHAM1_to_ASC_TF_TTL4CstoREFLWFSDC_TF_DuringHPIEXC.png: FINAL ANSWER PLOT 
      On the left panel, I show the *calibrated* transfer function from the table-top L4Cs to the REFL WFS DC "QPD" Pitch signals, in [m/m]. I know that audio-band "DC" spot displacement on the WFS isn't super interesting without further interpretation to, say, 
        - how much CHARD arm cavity motion that might confused for by the RF portion of the sensor, or
        - how much angular motion that would represent at RM1 or RM2,
      but that would be even more hard to calibrate. At least it's something. On the right, I mimic what I assume to be the "barely calibrated" LLO plot -- the WFS DC pitch signal in ADC counts, and the L4C signal in "ideal inertial sensor" counts, which are 1 [(nm/s)/ct] above 1-ish Hz, and the amplitude "dB" of all that, so, 20*log10 ( WFS DC Pitch [ct] / L4C [ct] ).

I also plotted the phase of the calibrated transfer function, just to see it. Interestingly, the phase between 6 and ~20-30 Hz is essentially zero. It really is direct coupling! What's surprising is that the 1e-8 [m_RMS] motion of the table top, which translates to 1e-10 [m_RMS] or 0.1 [nm_RMS] of spot motion, and that causes all these woes.

DETAILS
To calibrate the TT L4Cs, I found Hugh and I's aLOG from 2018 -- LHO:45501 -- which suggests that the individual L4Cs are calibrated into the the "usual" for the seismic group -- an ideal 1 Hz inertial sensor, that asymptotes to 1 [(nm/s) / ct] at high frequency (the L4Cs are in analog 'low' gain of 1.0 with no digital compensation for it in FM4, and the "conversion to ideal" Q adjustment filter is installed in the FM1 "Cal" module). Thus -- because the Z DOF is just the average of the three individual L4Cs, we can plug 
    Gain: 1e-09
    Poles: 0, 0
    Zeros: 0.707 0.707
in as the H1:HPI-HAM1_TT_L4C_Z_DQ channel to get units of [m/s], and then ask DTT to integrate once to get displacement in [m].

Calibrating the REFL WFS DC "QPD" signal is more difficult, but doable after phoning a few friends.
- Chiara DiFronzo wrote the most succinct and clear note I know of on how to calibrate QPDs, T1900105, which suggests through some approximations that
    dx / dP = (pi*w) / (sqrt(2)*P),
where dP is the change in power, dx is the change in displacement causing that change in power, w is the spot beam radius at the QPD, and P is the total power on the QPD.
For this, 
- Jenne pointed me to Elenna's recent work on confirming the calibration of the WFS DC path into Power at the PDs in [W] LHO:62976,
    dP [W] = pitch_ADC signal [ct] * standard 16 bit ADC gain * 1 / (trans impedance of 1000 Ohm) * quantum efficiency at 1064nm
    dP / dx [W/ct] = 40 / 2^16 [V/ct] * 1/1000 [A/V] * 0.8 [W/A]
    dP / dx [W/ct] = 7.6e-7 [W/ct]
- Craig helped me find Dan Brown's note about all the distances between PRM and the REFL WFS T1700227, and thankfully, Craig also had an email from Dan where he used that information to tabulate the beam radii at each A and B REFL WFS to be 
    w = 0.355 +/- 0.002 [mm] 
(the difference between the answer he gets for A and B is 0.002, of course there's no way the accuracy or precision of the number is good to the 2 [um] level).
- For the power on the photodiodes, P, I take
    - The power into the IMC, 2 [W],
    - Reduce it by power loss from the IMC to the PRM (the ratio of IMC-IM4_TRANS_NSUM_OUT_DQ / IMC_PWR_IN_OUT_DQ), 0.8 = 80%
    - Reduce it by the power reflection of the PRM (since the power transmission in is 3%, then that's) 1 - 0.03 = 0.97 = 97%
    - Reduce it by the amount of light that Craig's power budgeting (that we all know is questionable and one of things we'll validate when we go into HAM1) says: "only one part in 230 gets from the PRM to the REFL detectors," which I've rounded to 1/200 = 0.005 = 0.5%.
  So, that's
    P = Input power * IMC to PRM loss * PRM reflection * PRM to REFL PD loss
    P = 2.0 [W] * (40.6/50.3) * 0.97 * 0.005 
    P = 0.0078 [W] = 7.8 [mW]

Putting it all together, 
    dx / dP [m / adc ct] = 40 / 2^16 [V/ct] * (1/1000) [A/V] * (1/0.8) [W/A] * ( (pi * 0.355e-3) / (sqrt(2.0) * 0.0078) ) [m/W]
                         = 7.7e-08 [m/ct]
I can definitely believe that both the power on the photodiode, P, and the spot size at the QPD, w, are only accurate to a factor of 2, so I would say this calibration is only good to a factor of two, but that's good enough to make the point.