Previously in elog 8959, I gave a very simple method for determining the noise suppression behavior of the ISS. Recently, I recalculated this requirement in a more correct fashion and again redesigned the ISS to be used in the CTN experiment.
Just as before, the data from PSL elog 1270 is necessary to infer a noise suppression requirement. The data presented there by Evan consists of two noise spectra, 1) the unstabilized RIN presently observed in the CTN experiment readout and 2) the theoretical brownian noise produced by thermal processes in the mirror coating+substrate. The statement "TF_mag = (Unstabilized RIN) / (Calculated Brownian Noise Limit)", where TF_mag refers to the required open-loop gain of the ISS, is actually a first order approximation of the 'required' noise suppression. In fact if we wanted the laser noise to be suppressed below the calculated brownian noise level, it is more correct to say
Closed-loop ISS gain = (Calculated Brownian Noise Limit) / (Unstabilized RIN)
As this essentially gives a noise suppression spectrum i.e. a closed-loop gain in linear control theory. Below is a very simple block diagram showing how the ISS fits into the CTN experiment. The F(f) block represents my full servo board.
Some of the relevant quantities involved:
So looking at the block diagram, our full closed-loop transfer function is given by,
So then to determine the required F(f), i.e. the required transfer function for my servo, we consider the expression
The plant transfer function is simply Plant = (C(f) * a * P * A) ~ 0.014 V/V, where I have ignored the cavity pole around 97 kHz as our open-loop transfer function ends up crossing unity gain around 10 kHz. In the above, I have included what I call a 'safety factor' of 10. Essentially, I want to design my servo such that it suppresses noise well beyond what is actually required so that we can be sure noise contributions to experiment readouts are not significantly influenced by the laser intensity noise.
Using the data Evan reported for the brownian noise and free-running RIN, I came up with an F(f) to the meet the requirement as shown below.
Where the blue curve includes the safety factor mentioned before. This plot just demonstrates that using my modular ISS design, I can meet the given noise suppression requirements.
To be complete, I'll say a little more about the final design. As usual, the servo consists of three stages. The first is the usual LP filter that is always 'on' when the ISS loop is closed. The boosts I have chosen to use consist of an integrator with a single zero and a filter that looks somewhat like a de-whitening filter. The simulated open-loop transfer functions are shown below.