Summary:

I am inclined to believe that the arm cavity losses are such that the IFO is overcoupled. Some calculations, validated with Finesse modeling also suggest that there isn't a sign change for the CARM error signal when the IFO goes from being undercoupled to overcoupled, but I may have made a mistake here?

Details:

• We’d like to gain some insight into whether the interferometer is undercoupled, critically coupled, or overcoupled. Factors that determine which of these is true include:
  • Arm cavity losses
  • Recycling cavity losses
• The proxy by which we determine the recycling gain is usually the arm cavity transmission. Assuming T_PRM = 5.637 % according to the wiki, and assuming the arm cavity transmission is normalized to 1 when locked in the POX/POY state, we can say that the PRG is given by G_PRC = TRX × T_PRM, assuming that the (i) the RF sideband fields are perfectly rejected by the arm cavities and (ii) mode-matching efficiency between the input beam and the arm mode is the same as that between the input beam and the CARM mode.
• Apart from this, the other measurement we have available to us is the buildup of the sideband fields, namely POP22 and POP110. We can compare the values in the PRMI lock vs the PRFPMI to make some inference.
• I started off with an analytic calculation of the reflectivity of the compound arm cavity mirror.
  • Attachment #1 suggests we will have an over-coupled IFO for arm cavity losses below ~200 ppm, which is a regime we are almost certainly in now.
• Then, I repeat the analysis for the coupled CARM cavity, with the end mirror as the compound arm mirror and the input mirror as the PRM.
  • I assume 2 % loss in the PRC.
  • Attachment #2 shows that while the carrier field goes through a sign change in amplitude reflectivity (as expected), the sideband fields dont.
  • Per equation 4.2 of Koji's thesis, the error signal for CARM depends on the (signed) IFO reflectivity, and the absolute value of the derivative of the arm cavity reflectivity for the carrier w.r.t. CARM phase.
  • So, we don't expect the REFL11 signal to show a sign change.
  • The situation is more complicated for PRCL in REFL11, because as explicitly evaluated in Eq 4.3 of Koji's thesis, there are two terms that contribute, and their relative magnitudes will dictate the overall sign. 
• For a Finesse validation, I use a simplified 3 mirror coupled cavity to approximate the PRFPMI. I also retained the RF sidebands for diagnostic purposes. The idea was to study these PRG proxies and what their expected behavior is.
  • Attachment #3 shows the PDH error signal in the (arbitrarily defined) REFL11 I quadrature. While the optical gain changes as a function of the arm cavity loss, the actual slope does not change sign. The fact that the zero crossing doesn't happen at exactly 0 CARM offset is because of higher order mode light at the REFL port (in my model, I tried to preserve the flipped folding mirror situation so the mode matching between the arm cavity and PRC in my model is ~96%).
  • In fact, this may explain why a CARM_B offset is required to do the ALS-->IR handoff - the ALS servo wants to keep the arm offset to zero, but at that point, the PDH error signal isn't zero, and so the two loops end up fighting each other?
  • Attachment #4 is a more detailed study of the recycling gain as a function of arm cavity loss, but now including losses in the recycling cavity.

Conclusions:

1. I think the arm cavity losses are in the 60-80 ppm round-trip region. I don't see how we can explain the arm cavity transmission of ~350 otherwise.
2. The fact that REFLDC decreases as the arm transmission increases is because the input beam is getting better matched to the CARM mode, and there is less junk carrier light. 

Thoughts from others?