[Yehonathan, Yuta]
EDITED by YM on 22:11 June 27, 2022 to correct for a factor of two in the modulation index

Since we have measured optical gain in MICH to be an order of magnitude less compared with Yehonathan's FINESSE model (40m/16923), we measured the power at AS55 RF PD, and measured the modulation depths using Yarm cavity scan.
We found that 50/50 beam splitter which splits AS55 path into RF PD and RF QPD was not included in the FINESSE model. Measured modulation index were as follows:

TEM00 peak height: 0.6226 +/- 0.0237
RF11 peak height: 0.0067 +/- 0.0007
RF55 peak height: 0.0081 +/- 0.0014
RF11 modulation index: 0.208 +/- 0.012
RF55 modulation index: 0.229 +/- 0.020
RF11 modulation index: 0.104 +/- 0.006
RF55 modulation index: 0.114 +/- 0.010

Here, modulation depth m is defined in E=E_0*exp(i*(w*t+m*sin(w_m*t))), and m m/2 equals to square of the intensity ratio between sidebands and TEM00.

Power measurement at AS55 RF PD:
 - ITMY and ITMX single bounce reflection was measured to be 50-60 uW at the front of AS55 RFPD.
 - In the FINESSE model, it was expected to be ~110 uW with 0.8 W input to PRM (0.8 W * 5%(PRM) * 50%(BS) * 50%(BS) * 10%(SRM) * 10%(AS2) gives 100 uW)
 - In AP table, AS55 beam was split into two paths with 50/50 beam splitter, one for AS55 RF PD and one for AS WFS and AS110. This will be included in the FINESSE model.

Modulation depth measurement using Yarm cavity scan:
 - Aligned Yarm using ASS, and unlocked Yarm to get the 2sec scan data of C1:LSC-TRY_OUT_DQ, C1:LSC-POY11_I_ERR_DQ, C1:LSC-AS55_I_ERR_DQ.
 - TRY data was used to get TEM00 peak heights
 - POY11/AS55 data was used to find RF11/RF55 sideband peaks, and height was measured at TRY (see attached).
 - If we define m to be E=E_0*exp(i*(w*t+m*sin(w_m*t))), the amplitude of TEM00 I_00 is proportional to J_0(m) and the amplitude of upper/lower sideband I_f1 is proportional to J_1(m), where J_n(m) is the bessel function of the first kind.
 - m can be calculated using 2*sqrt(I_f1 / I_00).
 - Results were shown above. Error is calculated from the standard deviation of multiple measurements with multiple peaks,
 - The code for doing this lives in https://git.ligo.org/40m/measurements/-/blob/main/LSC/YARM/modulationIndex.ipynb

Discussion:
 - Power at AS55 account for the factor of 2, In the FINESSE model, modulation index of 0.3 was used (could be m=0.3/2 or m=0.3; needs check). These combined can explain a factor of 3 at least (or 6).
 - Gautam's measurement in Jan 2021 (40m/15769) gives almost double modulation index, but I'm not sure what is the definition Gautam used. It agrees with Gautam's measurement in Jan 2021.