I measured RIN -> frequency noise coupling coefficient at 10Hz from RCAV. The result is 0.02 Hz/uW (frequency shift/power fluctuation built up in the cavity*)

*by power fluctuation built up in the cavity, I mean [RIN x power input as measured at the input periscope x Finesse/ pi ]

Following the list I wrote down here, 

1) TF between FASTOUT/FASTMON is a lowpass with a pole at 10Hz. FASTOUT is fed to the laser for controlling its frequency. FASTMON is a channel for monitoring. This is as expected from the RC low pass filter at FASTOUT.

 2) I modulated the beam's polarization and used a PBS so the polarized transmitted beam is amplitude modulated. I used sine wave out from SR785, 2Vpp @ 10Hz connected to the EAOM. I measured the RCTRANSPD signal with an oscilloscope to check the signal is 211 mV +/- 7mV. The power is modualted by ~ 4%.

Then measure the peak at the modulating frequency (10Hz) from RCTRANSPD and FASTMON spectral density with SR785. The linewidth is 0.976 mHz. FASTMON was connected to chA, RCTRANSPD to chB. The data were averaged over 4 samples.

  The DC level of RCTRANSPD was 220 mV.

 The peak from FASTMON and RCTRANSPD are 6.75 mV/rtHz and 55.2 mV/rtHz. To convert this to the coupling coefficient we have to:

1) convert the voltage output from FASTMON to FASTOUT, the TF is measured and shown above. 

    V_fastout = V_fastmon x sqrt(  1 /   1 + (f/10Hz)^2  ), so  for 10Hz,

    V_fastout = V_fastmon x sqrt (1/2)

2) convert the Voltage out to frequency change by 3.09 MHz/V factor.

3) convert RCTRANSPD to RIN by dividing the PSD by DC level (220mV)

4) convert RIN to power fluctuation by x power input (1mW) x Finesse (9710) / pi

or

Freq/Hz   =  Vmon   x TF  x 3.09 [MHz/V]   

                   ------------------------------------------

                 V_RCtranspd / rctranspd_DC     x Pin  x Finesse/pi

  For 10Hz, 1mW power input, rctranspd_dc = 220 mV, we get

  RIN coupling coeff at 10Hz  =        0.0188 Hz/uW   = 18.8 mHz/uW.

I repeated the measurement with:

1)half modulating power, 1Vpp. the coupling coefficient decreases to  0.015 Hz/uW, so it seems that the modulation range I chose might be too large so the effect was not linear.  The coefficient should remain constant!!

2) 20Hz modulating power, 2Vpp (but the line width was 7.8 mHz, not 1mHz for quick measurement)

, the coupling coefficient is  9.35  mHz/uW. The result is smaller than that of 10Hz, which is expected from 1/f effect, but I think I should have used the same line width and look at a few more frequencies.

I'll have to check the coupling coefficient from ACAV, and check the beat signal to see if they are canceled or not.

Comparison to the calculation I did

The calculation I did here gives the noise at 10Hz from 10mW input, RIN = 10^-4, Finesse 10^4, to be 1.4 [mHz/rt Hz] which is already lower than the coating noise (10mHz/rtHz at 10Hz)

I can convert it to the estimated coupling coefficient to compare with what I measured.

1.4 mHz/rtHz       =   coupling coeff [Hz/W] x  Pin [W] x RIN [1/rtHz] xFinesse/pi,  or

coupling [Hz/W]    =   1.4 mHz/rtHz  / Pin [W]  / RIN [1/rtHz] / (Finesse/pi)

                            =   0.44 Hz/W

This is 5 order of magnitude lower than what I measured !!! and the calculated noise is only one order of magnitude lower than the coating. It means that the calculation is not correct and we cannot ignore the effect. we will run into it before reaching coating thermal noise if the effects on both cavities aren't canceled.