The PDH filter cavity signal has been calibrated injecting a line at 28 kHz (above the ugf ~ 10 kHz of the loop) on the “ramp” input of the electronic servo. The ramp input is summed to the PZT correction signal.

The amplitude of the 28 kHz line in Hz is obtained using the formula:

S_Hz = V_RMS  (V) * sqrt(2)*100*2e6 Hz/V = 1.25e-6*sqrt(2)*100*2e6 = 353 Hz 

Where V_RMS is the line amplitude measured by the Agilent spectrum analyser. The factor sqrt(2) is obtained to pass from the V_RMS to the line amplitude (the factor has been also experimentally verified looking the same line with the spectrum analyser and the oscilloscope).

The factor 100 is the reduction of the PZT_moni output . 2e6 Hz/V is the calibration of the PZT after the SHG.

Measuring the line at 28 kHz in the error signal and compensating for the cavity frequency pole is it possible to find the calibration factor K in V/Hz. The formula used is :

S_V  = K(V/Hz)* S_Hz /sqrt(1+ (f/f_0)^2)  

where f_0 = 1.5 kHz and S_V = sqrt(2)*38.9e-3 V 

--> K = 2.9e-3 V/Hz

which seems to be in agreement with the calibration obtained looking the PDH signal when the cavity is freely swinging. In that case we see a peak-to-peak of the PDH of ~ 4 V for 1.5 kHz of the cavity line which correspond to a K = 2.7e-3 V/Hz. Note that when the cavity is freely swining we have also rining effects which can perturb this measurement. 

We have also checked that reducing the frequency of the line sent to the PZT (with the same amplitude) to 14 kHz, the amplitude of the line of the error signal is multiplied by 2, as expected given the cavity pole. A more quantitative analysis (fully taking in account the effect of the loop) is necessary to check the position of the cavity pole.

Another test was to increase the amplitude of the line by a factor 10, thus having a 29 kHz line with amplitude of 3 kHz (two times the cavity width of the cavity). The cavity stays locked and the calibration factor measured is the same with the one measured with the line with an amplitude of 300 Hz. Increasing further the amplitude of the 28 kHz line to ~ 7 kHz (4 times the cavity linewidth) makes the lock more fragile, and sometimes the cavity unlocks. Moreover, an oscillation with a frequency of ~ 1 Hz appears in the error signal (but it is not accompanied with a similar oscillation in the transmitted power).