Alastair and I have been doing some thinking about servo loops, noise and readout schemes. Below is a simple servo diagram which highlights (most of) the important players.
P (plant): The cavity response in W/Hz (this contains the mixer demod and LPF implicitly). Really, P1 = P2.
D (diode): PD response in V/W. P and D are split up in this case so that we can add in shot noise in watts between them. Ideally, D1 = D2.
F (filter): the PDH box gains in V/V
A (actuator): the gains of the laser PZT (1) and AOM (2) in Hz/V.
δL: Laser frequency noise in Hz. This includes noise in the laser itself as well as mechanical noise in the beam path before the beamsplitter. Common to both loops.
δ1,2: Frequency noise imparted by mechanical effects between the beamsplitter and the cavity input mirror in the CCW and CW paths, respectively. Includes VCO frequency noise for CW beam. This noise is differential.
δc: Cavity noise from mirror motion in Hz. Common mode.
S: Unidirectional Sagnac frequency shift. Anti-common mode in the sense that its value in one loop is opposite that in the other. This is added in with δc because it is manifest as an apparent cavity length change.
SN1,2: Shot noise on REFL photodiodes in W. Differential.
δV1,2: Error signals input into the PDH boxes. The loops act to minimize these.
ER: The gyro signal in the reflection sensing scheme, which is just the actuation signal to the AOM.
ET: Gyro signal in the transmission sensing scheme. Mixer shown represents optical demod, have to include phaselock loop for electrical demod and frequency shift readout… working on this.
Conclusions that can be drawn so far:
ER is easy to read out, but contains extra noise. The signal fed from the CCW loop to the CW loop (this is just the laser PZT acting on the common laser) suppresses δL and δc, and it adds half the total Sagnac signal, S. However, it also imparts the shot noise of REFL CCW (SN1) and the CCW precavity frequency noise δ1. This means that ER is proportional to [2S + δ2 - δ1 + (1/P2)(SN2 - SN1)], that is, signal + differential precavity noise + differential shot noise suppressed by the cavity finesse.
In the transmission sensing scheme, the optical signals are combined after they have interacted with the input optics, and their differential noise is suppressed by the CW loop gain. They contain the frequency fluctuations required to stay matched to the cavity (-δc), but this noise is common mode and is rejected by the CMRR of the transmission demod. The only noise in the transmission scheme is thus the REFL shot noise suppressed by the cavity finesse as well as any other frequency noise occurring after the cavity mirrors (vibrations in the MZ, noise in the PLL VCO, and shot noise on the TRANS PDs, which is suppressed by a factor of f). If these can be attenuated to better than the junk at the input side, then we're better off in transmission.
Other notes: There is an additional nose source from phase modulation of the REFL signal (beat between leakage beam and SBs) caused by the mirrors on the way back through the faradays. These come in where SN does. Assuming small δϕ (which we can do because the mirrors are only moving a very small fraction of the beat wavelength), the resultant power fluctuation is second order in δϕ. Numbers coming soon but we think this can be ignored.
As far as the physical experiment is concerned, our next step is to monitor δV1 and adjust the CCW loop to minimize noise therein. The quieter this is, the better the common mode rejection. Then we can worry about gyro signals and sensitivity curves.