We have been explaining the excess noise in the gyro signal by saying that the residual common-mode noise that isn't removed by the CCW loop "spills over" into the CW loop, causing the CW servo to do what it can to ensure that the beam it controls is as closely matched to the cavity as possible. This effectively results in differential-mode noise, which of course shows up in the gyro signal.
It is important to understand what is meant by "common-mode noise". If the loops are ideal, then cavity length changes only affect the gyro signal insofar as they modulate "DC" signals, such as that from the earth's rotation or---more importantly---the 1-FSR shift between the two counter-propagating beams. When the loops are not ideal, the two servos don't do equally well at matching the optical frequencies of their respective beams to the cavity eigenmodes. This means that there is excess noise in the beat signal BEYOND the sagnac shift and the above fundamental noise couplings.
This is easy to understand in the case that the gyro is absolutely at rest and there is no FSR upshifting of the CW beam: in this case, ideal loops produce a null output, while non-ideal ones do not (in fact, all that matters in this case is that the CCW loop is ideal; if it is, then the CW error signal is null).
Below is the gyro noise/signal diagram that we made a few months ago. Miraculously, it is general enough that it remains accurate despite our vastly different understanding of how noise couples in. The point with the red dot is the point in some abstract space with units of frequency at which the CCW loop influences the CW one. If the CCW loop is ideal, and we ignore shot noise (which is low enough to be irrelevant), then the signal here contains (-1 x): the laser noise, the noise in the optics along the CCW path before the cavity, the common-mode cavity noise, and half the full sagnac shift. When this is summed into the CW servo, it eliminates the laser noise and the noise in all the optics before the beams are separated, as well as the common-mode cavity noise. What is left is the differential noise in the optical paths between the beam separation and the cavity and the full sagnac shift. Since we read out between F2 and A2, we also see the noise in A2 (the AOM and the VCO driving it).
If we allow the CCW loop to have non-infinite gain, then the ideally-equal-and-opposite signals are no longer nulled at the summation point, causing the CW loop to work harder than it needs to.
This mechanism has been informally verified by simply reducing the gain of the CCW PDH box and watching the gyro noise go up. A more rigorous test, which will also allow us to verify our displacement noise coupling model, is as follows:
- Take a spectrum of the CCW error signal. This is a measure of the residual common-mode noise.
- Calibrate (1) into frequency noise
- Take a spectrum of the CW actuation signal.
- Calibrate (3) into frequency noise
- Compare (1) and (3). We now know if the gyro signal is dominated by spillover noise.
- Calculate how much more gain we need in the CCW loop to reduce the spillover noise to below the level of the expected fundamental displacement noise couplings.
- Add this much gain.
- Repeat (1-5). Now, (3) should not go down if we increase the gain further. We will have reached fundamental noise limit for the gyro without length stabilization.