Arm cavity FWHM for IR is
FWHM = FSR / F = c/(2LF) = 8 kHz.
In cavity length, this is
L/f * FWHM = 40m/(c/1064nm) = 1.2 nm
So, to do mode scan nicely, arm length fluctuation during resonant peak crossing should be much less than 1.2 nm.
Let's consider only ADC noise and seismic noise.
* S: conversion from Y arm length to the beat frequency
dL/L = df/f
S = df/dL = f/L = c/532nm/40m = 1.4e7 MHz/m
* W: whitening filter
We set it to flat gain 50. So,
W = 50
* D: AD conversion of voltage to counts
D = 2^16counts/20V = 3300 counts/V
* B: frequency to voltage conversion of the beatbox.
We measured BWD(elog #6815). When we measured this, W was 10. So, the calibration factor at 0 crossing point(~ 50 MHz) is
B = 1400*0.048/10/D = 0.0021 V/MHz
* A: actuator transferfunction
I didn't measure this, but this should look like a simple pendulum with ~ 1 Hz resonant frequency.
* n_ADC: ADC noise
ADC noise is about
n_ADC = sqrt(2*LSB^2*Ts) = sqrt(2*(20V/2^14)**2*1/64KHz) = 1.6 uV/rtHz
* n_seis: seismic noise
We measured this by measuring C1:ALS-BEATY_COARSE_I_IN1. This is actually measuring
D(WBSn_seis + n_ADC)
Calibrated plot is the red spectrum below.
* F: servo filter (basically C1:ALS-YARM)
We need to design this. Stabilized arm length fluctuation is
x_stab = 1/(1+G)*n_seis + G/(1+G)*n_ADC/(WBS)
where openloop transferfunction G = SBWDFA.
Below ~ 50 Hz, n_seis is bigger than n_ADC/(WBS). We don't want to introduce ADC noise to the arm. So, UGF should be around 50 Hz. So, we need phase margin around 50 Hz.
We also need about 10^3 DC gain to get the first term comparable to the second term.
Considering these things, openloop transferfunction should look like the below left. Expected error signal when ALS on is the below right. I put some resonant gain to get rid of the peaks which contribute to the RMS (stack at 3.2Hz, bounce at 16.5 Hz).
Inloop RMS we get is about 0.3 nm, which is only 4 times smaller than FWHM.
We need to reduce RMS more by factor of ~ 30 to get resolusion 1% of FWHM.
Most contributing factor to the RMS is power line noise. We might want comb filters, but it's difficult because UGF is at around this region.
So, I think we need more fancy whitening filters. Currently, we can't increase the gain of the whitening filter because SR560 is almost over loading. Whitening filter with zero at 1 Hz might help.