I calibrated PRM actuation transfer function measured in alog #10450.
Measured PRY error signal is smaller by factor of 2 from the calculation and suspension model. This means that demodulation phase is off by 60 deg, or PRY modematch(including misalignment) is 50%, or suspension model is off by factor of 2 (or combination of all of them).

[Motivation]
We wanted to check the PRCL loop signal chain (We have done this for MICH loop already; see alog #10213).
Also, we need calibrated actuation TF for designing the compensation filter which does not saturate DAC.

[Method]
1. Made PRY simulink model (It lives in /ligo/svncommon/NbSVN/aligonoisebudget/trunk/PRMI/H1).

2. Change optical gain from PRM motion to REFLAIR_A_RF45_I to match the measured OLTF (which was measured in alog #10450).

3. Use this optical gain to calibrate PRM actuation transfer function.


[Result]
1. OLTF_PRCL_1077847156.png: OLTF compared with model and measured. Flat gain is fitted in the model and this gives the optical gain. The measured optical gain was 1.3e3 W/m.

2. From the REFLAIR signal chain in alog #10213, calibration factor for REFLAIR_A_RF45_I_ERR in PRY is 3.4e11 counts/m.

3. ActTF_PRM_1077847156.png: Calibrated PRM actuation transfer function. Red curve is plotted using zpk from LISO fitting of the measured TF (alog #10450) and divided by 3.4e11 counts/m for calibration. Blue/Cyan curve is from the suspension model using /ligo/svncommon/SusSVN/sus/trunk/Common/MatlabTools/TripleModel_Production/generate_Triple_Model_Production.m and calibrated using the numbers from ./MatlabTools/make_OSEM_filter_model.m (or LIGO-T1000061). M3 and M2 crossover and measurement look healthy. Note that the overall gain of the measurement agrees with model just because we don't have independent measurement of the optical gain. Even so, crossover frequency doesn't change.


[Discussion on optical gain]
Theoretical expression for PDH signal is

dPmod/dL = 2*8*pi/lambda*Peff*J0(beta)*J1(beta)*(t1**2*r2)/(1-r1*r2)

With

Effective input power: Peff = 7.3 uW * 4 /Tprm**2 = 0.032 W  (alog #10213; incident power on REFLAIR_A was 7.4uW when PRM and ITMY is misaligned)
Modulation depth beta=0.07 (alog #9395)
Amplitude reflectivity/transmissivity of PRM: t1 = sqrt(0.03)
Amplitude reflectivity of BS/ITMY compound: r2 = rBS*rBS*rITMY = 0.50

This gives dPmod/dL = 3.1e3 W/m (+/- ~10%). Here, Pmod is RF modulation amplitude of laser power, and dL is one-way length change of PRC, which equals to PRM motion. (Optickle gives 1.5e3 W/m since Optickle assumes demodulation gain of 1/2).

Even if I include the loss of the cable we measured(alog #10213), theoretical value is 2.5e3 W/m (= 3.1e3 W/m * 0.81), or 6.5e11 counts/m at I_ERR. This is factor of 2 larger than the measured.

Since theoretical value assumes perfect modematching and demodulation phase, actual value might be smaller. Also, note that measured optical gain is derived from the model which assumes that suspension model is acurate enough.

[How to solve this challenge]
 - Calibrate BS actuation transfer function using simple Michelson, and compare it using the measurement done in PRY. This will be an independent measurement of PRY optical gain.
 - Measure PRY modematching