Arm cavity round-trip loss and changes in ITM reflectivity were estimated using measured power recycling gains and arm cavity finesse.
Increase in the optical loss inside the PRC from 2% to 10%, and increase in the ITM transmission by ~500ppm can explain both power recycling gain and finesse measurements.
These suggest that power recycling gain reductions are mainly from PRC loss increase, and arm cavity finesse reductions are mainly from ITM transmission increase.
Arm cavity round-trip loss (including ETM transmission) stays almost unchanged within ~20ppm.

Assumptions:
  - Nominal mirror transmissions are the ones summarized in klog #36231, but losses inside the PRC (including ITM AR losses), ITM HR transmission and arm round-trip loss changes over time.
  - Losses in PRC (TPRC) are the same for carrier and sidebands. NOTE that this could be wrong due to, e.g., Lawrence effect.

What we did:
  - Used the same data set as klog #36231.

  - This time, sideband power recycling gain was estimated using K1:LSC-POP_PDA2_RF90_I_ERR_DQ, divided by K1:LAS-POW_IMC_DC_INMON to normalize for the input power changes (see left bottom plot). Overall factor was calibrated by calculating the nominal sideband PRG at the beginning of the data using the following equations. TPRCorig is the nominal losses in PRC, and was eyeball fitted to be 2% to have the arm cavity round-trip loss result consistent with klog #30823 (see right middle plot).
tp=sqrt(T['PRM'])
rp=sqrt(1-T['PRM'])
ranti=sqrt(1-TPRCorig)
PRGsideband=(tp/(1-rp*ranti))**2

  - Using the power recycling gain for sideband and carrier,  A=Tarm+TPRC (see red and blue dots in right bottom plot) and B=Tanti+TPRC (see yellow dots in right bottom plot) were estimated, where Tanti is the loss from the arm for sidebands, TPRC is the losses in PRC, and Tarm is the loss from the arm for carrier. Since estimated Tanti+TPRC is huge compared with TITM, it is safe to assume that B is mostly from TPRC.

  - Therefore, A-B gives Tarm (see cyan and magenta dots in right bottom plot). Note that in this calculation, we are assuming that TPRC is the same between carrier and sideband. Changes in A B (EDITED ON Jan 28) is actually mostly from TPRC, and Tarm is roughly constant over time. This means that the arm cavity round-trip loss is roughly constant over time.
 
  - Using the following simultaneous equations, TRTL and Tarm can be measured, where TRTL is the arm cavity round-trip loss. Results are plotted as cyan and magenta crosses and stars in right middle plot.

A-B = Tarm = TRTL*4/TITM
Finesse = 2*pi/(TITM+TRTL)

Discussion:
  - Assumption that TPRC is 2% initially is just a rough guess (+/- 0.5% or so to make TRTL consistent with klog #30823), and it does not change the overall trend very much. If POP90 is properly calibrated, this will not be necessary.
  - Huge increase in the PRC loss TPRC to 10% is a mystery. TPRC includes ITM AR losses and the Michelson fringe conditions.
  - If TPRC is smaller for carrier than that for sidebands due to, e.g., Lawrence effect, TRTL increases and changes for TITM reduces. This is also a plausible senario.

Next:
  - Redo this calculation once the fitting for the finesse measurements are fixed by Oshino-san.
  - Investigate the cause of huge increase in the PRC loss (AR frosting? Resonant condition changes for p-pol? Birefringence changes? Michelson contrast changes?)
  - See if ITM reflectivity changes are consistent with lce layers (see, also, klog #34903)
  - Calibrate POP90 using modulation depths and power at the PD so that we don't need to guess nominal losses in PRC.

In the previous study, I assumed that the losses in PRC (TPRC) are the same for carrier and sidebands.
If this is different due to, e.g., Michelson Schnupp asymmetry and Lawrence effect, estimated PRC losses for sidebands will be different.
To demonstrate this, I have plotted the case where initial PRC loss for sidebands being 10% (roughly the same as single bounce loss to p-pol due to ITM birefringence; JGW-G1910369) and PRC loss for carrier being 0.2 of that for sidebands (this gives initial loss of 2% for carrier, to get the consistent arm cavity round-trip loss with klog #30823).
In this scenario, Tarm was estimated using A-0.2*B.

This also well explains the measured arm cavity finesse drop and power recycling gain drop.
Increase in the optical loss inside the PRC will be 10% to 23% (can be anything), and increase in the ITM transmission stays to be ~500ppm.
Estimate of the increase in the ITM transmission is robust since we cannot attribute the main cause of the finesse drop to round-trip loss increase (see klog #36231).

This senario may be the most plausible scenario.

Next:
 - Calibrate POP90 with expected sideband power recycling gain for PRX (or PRY). For PRX or PRY, expected sideband PRGs are dominated by losses from BS, and are the following.

Gprx=(tp/(1-rp*tBSs*tBSs))**2
Gpry=(tp/(1-rp*rBSs*rBSs))**2
PRX sideband PRG 0.400 +/- 0.004
PRY sideband PRG 0.344 +/- 0.003

Here, I used the following measured BS transmission and reflectivity (klog #29284).

tBSs**2=51.9 +/- 0.2 (stat.) +/- 0.2 (sys.) # for s-pol
rBSs**2=47.7 +/- 0.2 (stat.) +/- 0.2 (sys.) # for s-pol

Now that we have calibrated POP90 into PRG for f2 (klog #36340), and have updated finesse results (klog #36297), I redid the estimate of the losses in the IFO.

The loss in PRC probed by f2 has increased from ~10% in March 2025 to ~25% in November 2025.
(These numbers do not incude the losses from f2 detuing in PRC and Schnupp asymmetry mismatch to f2.)
Assuming that loss in PRC probed by carrier is smaller by a factor of ~6 (consistent with measured Lawrence effect klog #36283), round-trip loss in the arm cavity has increased from ~80 ppm to ~100 ppm during O4abc.
The drop in the finesse during cryogenic operation is mostly due to ITM transmission increase.

Method:
 - Assume PRM transmission T_PRM is constant.
 - As described previously (klog #36233), T_RTL*4/T_ITM+T_PRCc can be estimated from normalized arm cavity transmission. T_PRCs can be estimated from the power recycling gain for f2, as measured by POP90. Here, T_RTL is arm cavity round-trip loss (including ETM transmission), T_ITM is ITM transmission, and T_PRCc and T_PRCs are PRC loss for carrier and sideband, respectively.
 - First, POP90 sqrt(I^2+Q^2)/Pin was multiplied by 1.88(9) (see klog #36340) to get f2 PRG (bottom left, blue dots). The value was corrected into non-detuned value by multiplying the PRG with (1+(det/HWHMprc)**2). From the corrected PRG value, T_PRCs=1-rloss**2 can be estiated because PRG = tp/(1-rp*rmi*rloss). Note that rmi**2=0.97445(15), so the loss from Schnupp asymmetry mismatch is ~3% and is non-negligible. See bottom right, yellow dots for the estimated T_PRCs. As a sanity check, the estimated value is consistent with recent measurements (klog #36340).
 - T_PRCc cannot be measured independently from arm cavity transmission. Assuming that the PRC losses are mainly from birefringence, T_PRCc can be guessed from T_PRCs. Assuming T_PRCc = T_PRCs / 6(2), T_RTL*4/T_ITM can be estimated from normalized arm cavity transmission (bottom right, cyan and megenta).
 - Finesse measurements (top right) gives you 2*pi/(T_RTL+T_ITM), so by combining the two, T_RTL and T_ITM can be estimated (middle right).
 - The assumed ratio T_PRCs/T_PRCc=6(2) comes from the amount of reduction in p-pol observed for ITM single bounce and arm locked (klog #36283). This might not be true if the loss is not dominated by birefringence. But this ratio gives the estimated T_RTL in March 2025 consistent with the measured T_RTL in August 2024 (klog #30823), so it seems not to far from what's happening.

Discussion:
 - We have time-varying T_ITM, T_RTL, T_PRCc, T_PRCs. But we measure only have arm cavity transmission, POP90 and occasional finesse measurements. If we can also occasionally measure T_RTL with the method described in klog #36283, we can estimate the 4 parameters independently. We should have measured T_RTL at least once at cryogenic temperatures.
 - The reason why the loss in PRC probed by f2 has increased from ~10% to ~25% is unclear. Increase by ~15% is not seen by carrier, so it is not a simple loss common to f2 and carrrier. It could be due to increased birefringence at cryogenic temperatures or birefringence in ice layers?
 - Updated finesse values (klog #36297) seems to be lower by ~50.

Next:
 - Make a script to measure arm cavity round-trip loss occasionally
 - Check if the rate of ITM transmission change is consistent with ice layer formation