Koji asked me to perform a simulation of the response of POP QPD DC signal to mirror motions, as a function of the CARM offset. Later than promised, here are the first round of results.
I simulated a double cavity, and the PRC is folded with parameters close to the 40m configuration. POP is extracted in transmission of PR2 (1ppm, forward beam). For the moment I just placed the QPD one meter from PR2, if needed we can adjust the Gouy phase. There are two QPDs in the simulation: one senses all the field coming out in POP, the other one is filtered to sense only the contribution from the carrier field. The difference can be used to compute what a POP_2F_QPD would sense. All mirrors are moved at 1 Hz and the QPD signals are simulated:
This shows the signal on the POP QPD when all fields (carrier and 55 MHz sidebands) are sensed. This is what a real DC QPD will see. As expected at low offset ETM is dominant, while at large offset the PRC mirrors are dominant. It's interesting to note that for any mirror, there is one offset where the signal disappears.
This is the contribution coming only from the carrier. This is what an ideal QPD with an optical low pass will sense. The contribution from the carrier increases with decreasing offset, as expected since there is more power.
Finally, this is what a 2F QPD will sense. The contribution is always dominated by the PRC mirrors, and the ETM is negligible.
The zeros in the real QPD signal is clearly coming from a cancellation of the contributions from carrier and sidebands.
The code is attached.
const Pin 1 # input power
const Lprc 6.752 # power recycling cavity length
const d_BS_PR3 0.401 # folding mirror distances
const d_PR2_PR3 2.081
const d_PRM_PR2 1.876
const c 299792458 # speed of light
const fmod 5*c/(4*Lprc) # modulation frequency, matched to Lprc