= Background =
See 9928.
The RMS of the ALS sensing noise was limited by test mass motion in L direction at 0.15 Hz.
However, naively thinking, arm length fluctuation could not be a source of the sensing noise of the ALS system since arm length fluctuation would be suppressed by the control by the ALS if the gain of the control is high enough.

= Effect of the IMC cavity pole on the performance of the ALS =
The naive explanation does not hold when the IMC is taken into account.
See the following slide (P15 of JGW-G1910211).
What is controlled by the ALS and what is injected to the main IFO are slightly different due to the existence of the IMC.
This leads to the suppression limit of the ALS.

= Results =
I overlapped the spectra of the sensing noise of the ALS and the suppression limit of the ALS derived by the formula described above.
Signals of LS oplevs of ITMX and ETMX are used to get the spectrum of the arm length fluctuation.
I used 5.2 kHz as the value of the cavity pole of the IMC (7671).
The coherence between the sensing noise and the arm length fluctuation is also plotted.
We can see that these two spectra coincide well around 0.15Hz!!

= Note =
- This result indicates two things. One is we have to damp the test masses to achieve better performance of the ALS even in noisy days
The other is we have to consider picking off the transmission of the IMC for the PLL of the ALS in a future detecotr.
- Kozapy gave us good example codes of GWpy so that I could generate these plot easily. Thanks