Evan, Kiwamu, Stefan

Today we tried to measure the PRC cavity length using the single-shot time-delay method. Despite an estimated accuracy of about 2mm, we only achieved a precision of about 1cm variation between measurements.

Measurement details:
- We put a 3400Hz, 7.5V (25000cts using LSC-EXTRA) signal into the IMC offset.
- We intentionally detuned the modulation frequency to generate maximal AM at 3400Hz in the mode cleaner.
- We then demodulated this signal after a single bounce off the PRM, ITMX and ITMY respectively to get a time-of-flight measurement.
- Before we started we made sure the REFLAIR demodulation was exactly 90deg (with an error of about 1e-3deg), and the relative gain was balanced to 1 part in 1e4. We achieved this by adding a 45MHz band-pass into the PD RF path of the demodulator (to get rid of any higher harmonics contributions), and then adding a test signal exactly 3.4kHz offset. Using this test signal we balanced the gain and phase response. All this was done at a fixed whitening gain. (The 45MHz band-pass was taken out again at the end of the measurement.)
- With this setup we achieved a phasing precision for PRM of about 0.02deg, and about 0.1deg for the IMTs. However there were large q-offsets, so we had to be careful to tune the audio-demodulation phase for the 3.4kHz signal. Probably our biggest systematic error came from this.
- From the measured phase difference we can calculate the expected PRX length (and PRY and Schnupp length in the same way) using the following script:

MATLAB script to calculate the length:
f=9099483; % sideband frequency
c=299792459;
lambda=c/5/f;
degX=-9.356;    % demodulation phase for ITMX
degPRM=-10.311; % demodulation phase for PRM
dX=degPRM-degX;
Lx=(17.5+dX/360)*lambda/2

- To verify the result, we then calculated the frequency which should be exactly anti-resonant, and repeated the measurement. Unfortunately the results only were good to about 1deg in phase difference, or about 1cm:
      f (Hz)  PRM phase   ITMX phase   PRx length   new f (Hz)
1st) 9101734   73.87deg    72.311deg    57.6557m     9099483
2nd) 9099483  -10.311deg   -9.356deg    57.6469m     9100863

The two results disagree my 8.8mm. While we did measure PRY too, we had different whitening gain settings for PRM and ITMY, so we didn't trust those numbers.

We did however measured the Schnupp-asymmetry using the same technique: we got 8.9cm for Lx-Ly. Again, this number probably has about 1cm of systematic error.

For reference, the design PRC length is 57.6557m (T1000298). With an 8cm Schnupp-asymmetry we expect PRx to be 57.6957m.