Vicky, Jeff, Dhruva

Today we made an attempt to project FC length noise to back-scatter in DARM. Here is the result and the steps followed to get to it. The following procedure is very rough and possibly inaccurate to various factors of ~2 in places, but hopefully sets up a procedure to do this more carefully in the future. 

FC IR Length Noise : 

From the SQZ data quiet time, I got a spectrum of the REFL_WFS_A_I_SUM in counts. 

To calibrate this to Hz, the naive idea is that we measure the peak-peak amplitude of the error signal and call that a FWHM of the cavity. However, some on doing some CLF-RLF signal modelling (that I don't trust completely yet), it looks like this method over-estimates the length noise in the cavity by a factor of around 1.8. 

The peak-peak fluctuation in I with the FC unlocked was around 14000 counts and the FWHM of the cavity using the ADF estimated parameters is around 74Hz. This gives us a calibration factor of (74/14000 * 1.8 * 300/280e12 *1e12) from counts to pm, with the last few factors being a conversion from Hz to pm. With this, the in-loop 'calibrated length noise' of the FC looks like this.

In hindsight, it would have been good to measure a new FC-IR open loop transfer function for the configuration when the data was taken but this opportunity was missed this time. We had to use an older transfer function, measured on January 24th.

In order to estimate the true length noise of the FC, I removed the loop suppression from the FC-IR signal to get the environmental plus sensing noise (the gain of the FC-IR loop was adjusted by a factor of 0.75 to make the high frequency noise look flat, this might roughly account for most of the IR loop difference between now and last month). Using the flat noise as sensing noise, I applied the loop suppression to the noises again and reinjected the sensing noise with the closed loop gain.

This is the 'true length noise of the FC calibrated to units of picometers'

FC Length to DARM Backscatter :

Very very rough procedure. In the absence of a locked IFO to make new and careful measurements on, the only available resource was this log by Daniel from December where he measured the FC backscatter after injecting a length excitation to the M3 stage of FC2. I tried to use this to get a rough conversion of FC length in pm to backscatter in DARM m. 

From the plot in the log, it looks like the rms value of the M3_L line is 200*sqrt(0.187) counts. With Jeff's help, I was able to roughly convert this to m in FC2.

L_rms_fc  = counts_rms*M3_LOCK_GAIN@40Hz * counts2V(DAC) * V2A(coil drivers) * A2N(magnets) * N2m(HSTS dynamics)

RMS_L_counts = 200*np.sqrt(0.187)
M3_LOCK_GAIN = 2.78
counts2V = 20/(2**18)
V2A = 0.326e-3
A2N = 2.81e-3
N2L = 5.43e-6

which gives L_rms_fc ~ 0.09 pm

According to the Daniel, the backscatter in DARM within a 2.5Hz bandwidth of the line is around 5 times the ambient level. This puts it at around 6.8e-20 m. I used a newer darm reference to calculate the ambient level. The noise isn't too far of from where it was in December. 

Using these two numbers, we can get a rough conversion factor from FC length noise to DARM scatter ~7.5e-19m/pm. 

Multiplying this with the FC length noise gives us the projection to DARM. It looks like this roughly calculated backscatter is below DARM but not by a lot at ~100Hz. This might also be because of the inaccuracy in the loop inversion and noise estimation in that frequency. Otherwise it looks like the backscatter is mostly below 1e-20m/rtHz.