I kept looking into the IRMC locking issue. I decided to start by splitting the measurement of the control loop transfer function

Figure 1 shows a model of the control loop for the electronics boards that we use in TAMA (Eleonora Capocasa thesis appendix D). We can extract the following transfer functions using noise injection and frequency response measurement:

• Servo transfer function: Noise to PERTURB IN, Frequency Response PZTmon/EPS2 - This is the transfer function of the electronics components inside the circuit board. Generically, it has an integrator response at low frequency (can set to single integrator 1/F or triple integrator 1/F^3), is flat or low slope across the UGF of the cavity in question, then rolls off at high frequency.
• "Optomechanical" transfer function: Noise to RAMP, Frequency Response EPS2/PZTmon - This is the transfer function of the light going in and out of the cavity when the piezo is excited. It should be generally flat until the cutoff frequency, i.e. low pass filter. The noise injection to the piezo will excite the mechanical resonances of the piezo and invar spacer, so it becomes a low pass filter with mechanical peaks. Actually, I don't really like this "optomechanical" terminology, to me it means 3 different things: 1) Thorlabs' terminology for moving mounts of optics such as translation stages, 2) The effect of spurious mechanical vibration of tables and housings on the amount of power going in and out of a cavity, 3) The direct interaction of quantum radiation pressure with moving mirrors (Braginsky and Khalili yellow book, etc).
• "Open loop" transfer function: Noise to PERTURB IN, Frequency Response EPS1/EPS2 - This is the open loop transfer function of the controlled cavity on lock. But again, I don't like how it is often generically called "open loop transfer function", because the above two are also open loop transfer functions

I measured the open loop transfer function for the optical cavity, as well as the electronics. The cavity transfer function looks fine (figure 2 - note that the appropriate noise excitation level is 100x lower when injecting to RAMP vs PERTURB IN). Basically the same as the reference version in the wiki. The electronics transfer function (figure 3, 4) is very low though. It has a -20dB/decade slope across the supposed UGF. I adjusted the gain of the servo and it didn't make much difference in the shape. Looking at the shape a bit more, I figured that it might be an issue with the integrator, since it is missing a lot of low frequency gain. So I tried to do the measurement again switching to 1/F^3 integrator, but the IRMC mostly refused to lock even when I tried different gain settings. I did see a bit of a flash of transmission (servo gain = 4) so I don't think it is just the 1/F^3 switch. Maybe it is something else along the chain that is badly behaved when in the presence of a triple integrator.

Some reference curves for the GRMC are shown in figure 5, from Yuhang's thesis. These are for the GRMC, but it has the exact same geometry as the IRMC. The overall unity gain frequency of the loop is 2 kHz. The servofilter has a -10 dB/dec slope across this frequency. The servofilter itself has a UGF of 200 Hz and a stronger slope at lower frequency. by comparison, the IRMC servofilter pretty much just strongly suppresses the signal even at 10 Hz.

Next, I tried looking at the RF sideband level. For the EOM used in the IRMC locking (QUBIG PM8-NIR_88), we have the following parameters:

• Resonant frequency: 88.3 MHz
• Applied signal (DDS1 channel 0): Output -8.5 dBm -> + 14.1 dB (RF amplifier board at bottom of FC cleanroom electronics) -> 5.6 dBm 
• Modulation depth at 1064 nm, 5.6 dBm: 0.2 rad (guess based one some generic datasheet, I don't have the actual one)

Using these, I estimated the amplitude of 88 MHz sidebands applied to the beam, with the following relevant parameters:

• 1064 nm photoelectric response: 0.6 A/W
• Transimpedance gain (50 Ohm load): 5.0 x 10^3 V/A
• -3 dB bandwidth: 150 MHz

If the bandwidth specification actually means a linewidth of 75 MHz, then having the 88 MHz sidebands out of the linewidth would result in an extra -2.7 dB attenuation past the 3 dB point (assuming first order rolloff for the photodiode), i.e. -5.7 dB power. Using the transimpedance gain there is 3.0 V/mW from optical power to PD signal. 

I measured the RF sidebands directly from IRMC REF RF (i.e. the cable from the PD that is filtered with a DC block) with the IRMC in scan mode, applying a T before the electronic signal goes to the mixer with the demodulation signal from DDS1. The sidebands have a level of approximately 60 mV. This implies a power level of 0.02 mW. The sideband relative intensity at 0.2 rad modulation depth is J_1(0.2) = 0.1. We start with 1.68 mW x J_1(0.2), then there is an extra -5.7 dB attenuation = x 0.27, and then divide again by 2 for 2 sidebands. This gives 0.023 mW converted by the PD into voltage for one first order sideband. So it seems consistent with us having -5.7 dB attenuation of sidebands at the PD due to being out of band.