I calculated a noise budget for the MICH using AS55 as a sensor. The calculation includes closed-loop TF calculations.
The notebook and associated files can be found on https://git.ligo.org/40m/bhd/-/blob/master/controls/compute_MICH_noisebudget.ipynb.
Attachment 1 shows the loop diagram I was using. The equation describing the steady-state of the loop is
![\left[\mathbb{I}-G \right]\begin{pmatrix} \gamma \\ \delta \\ \Delta\end{pmatrix} = \begin{pmatrix} \alpha \\ \beta \\ \epsilon\end{pmatrix}](https://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cmathbb%7BI%7D-G%20%5Cright%5D%5Cbegin%7Bpmatrix%7D%20%5Cgamma%20%5C%5C%20%5Cdelta%20%5C%5C%20%5CDelta%5Cend%7Bpmatrix%7D%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Calpha%20%5C%5C%20%5Cbeta%20%5C%5C%20%5Cepsilon%5Cend%7Bpmatrix%7D)
, where G is the adjacency matrix given by
First, the adjacency matrix G is constructed by stitching the small ABCDE matrices together. Once the inverse of (I-G) is calculated we can simply propagate any noise source to  and then calculate ![\left[\mathbb{I}-E(CA+DB)\right]B^{-1}\delta](https://latex.codecogs.com/gif.latex?%5Cleft%5B%5Cmathbb%7BI%7D-E%28CA+DB%29%5Cright%5DB%5E%7B-1%7D%5Cdelta) to estimate the displacement of the optics.
Attachment 2 shows the calculated noise budget together with Yuta's measurement.
All the input and output electronics are clumped together for now. Laser noise is irrelevant as this is a heterodyne measurement at 55MHz.
It seems like there is some mismatch in the calibration of the optical gain between the measurement and model. The missing noise at 3-30Hz could be due to angle-to-length coupling which I haven't included in the model. |