Measurement
Looking back at what I did in april (see log #8411) I realized that it is possible to get an estimate of how much the PRC length is wrong looking at the splitting of the sideband resonant peak as visible in the POP_110_I signal. With the help of Jenne the PRMI was aligned and left swinging. The first plot shows a typical example of the peak splitting of 55MHz sidebands. This is much larger than what was observed in April.
When the sidebands resonate inside the PRC they get a differential dephasing given by
dPhi = 4*pi*f_mod/c * dL
where dL is the cavity length error with respect to the one that makes the sidebands perfectly resonant when the arms are not there. This is not exactly the error we are interested in, since we should take into account the shift from antiresonance of the SBs in the arm cavities.
Nevertheless, I can measure the splitting of the peak in units of the peaks full width at half maximum (FWHM). I did this fitting few peaks with the sum of two Airy peaks. Here is an example of the result
The average splitting is 1.8 times the FWHM. Knowing the PRC finesse, one can determine the length error:
dL = c / (4 * f_mod * Finesse) * (dPhi / FWHM)
Assuming a finesse of 60, I get a length error of 4 cm.
To get another estimate, we kicked the PRM in order to get some almost linear sweeps of the PRC length. Here is one of the best results:
The distance between consecutive peaks is the free spectral range (FSR) of the PRC cavity. Again, I can measure the peak splitting in units of the FSR and determine the length error:
dL = c / (4 * f_mod) * (dPhi / FSR)
The result is again a length error of 4 cm.
Simulation
An error of 4 cm seems pretty big. Therefore I set up a quick simulation with MIST to check if this makes sense. Indeed, if I simulate a PRMI with the 40m parameters and move the PRC length from the optimal one, I get the following result for POP_110_I, which is consistent with the measurement.
Therefore, we can quite confidently assume that the PRC is off by 4 cm with respect to the position that would make the 55 MHz sideband resonant without arms. Unfortunately, it is not possible with this technique to infere the direction of the error.
