[Evan, Paul]

Here is a summary of the analysis of beam size measurements reported in 9898 and previous, in context with expected parameters from a model of the PRMI.

The first attached plot shows the ITMX direct reflection : PRM direct reflection beam size ratios predicted at the ISCT1 measurement location, as a function of ITMX susbtrate lens and PR2-PR3 distance offset*. The diagonal lines on the plots indicate the measured value of this beam size ratio**. There are also 3 pairs of red and blue vertical lines on the plots. The blue lines represent "cold states" and the red lines represent the designed for "warm state" with a +50km (+20uD) thermal lens in the ITM. The left-most pair of blue and red lines corresponds to the "expected" ITM lens state: -80km (-12.5uD) non-thermal lens in the ITMX substrate. The middle pair of blue and red lines corresponds to the "design" ITM lens state: no non-thermal lens in the ITMX susbtrate. The right-hand pair of blue and red lines corresponds to the "estimated" ITMX lens state, given the assumption that the generalized PRC parameter 2*PR2->PR3 - PR3Rc is as designed.

By assuming PR2-PR3 is as designed, we can estimate an ITMX non-thermal lens of +15uD (6.7km). The estimate from x-axis data and y-axis data agrees very well here. By assuming that the ITMX non-thermal lens is the -12.5uD (-80km) that came from surface figure measurements, we can estimate that the PR2-PR3 offset is +8.82mm. However, it should be noted that other simulation results showed that PRY is very likely to be unstable for such a large PR2-PR3 offset. As far as I'm aware, this was not observed during PRMI commissioning, though I'd appreciate some confirmation if anyone has any. Of course, any combination of PR2-PR3 offset and ITMX lens deviation that gives beam size ratios along the diagonal line is not discounted by these measurements.

The second attaced plots shows the same data for the ITMY measurements, which were taken while 4W was being applied to the ITMY ring heater. For ITMY we have no prior information about the non-thermal substrate lens, so the blue and red lines are not added. The only sensible prior information to assume, just for comparison, is that the non-thermal substrate lens is 0uD (+inf km). This is the right hand edge of the plot. The black dashed line represents the expected substrate lens caused by 4W of heating with the ring heater, using Aidan's number of -13.6uD/W. The diagonal lines again represent the measured beam size ratios. This seems to suggest that the non-thermal substrate lens is around +32uD (31.25km) in the x-axis and +40uD (25km) in the y-axis, again under the assumption that PR2-PR3 distance is as designed. From the ITMX plot and the ITMY plot, we should at least be able to pin down the difference in non-thermal lens power between ITMX and ITMY, even if we can't pin them down individually.

Next, I wanted to take a look at the consequences for actual mode matching in the interferometer, specifically between the IMC and PRX, PRX and XARM.
The third and fourth attached plots shows the IMC-PRX eigenmode overlap and PRX-XARM overlaps respectively over a slightly larger xaxis-range than for the first attached plot, but with the same blue and red lines for ITMX susbtrate lenses. Both diagonal lines from the plots in the first attached figure are included on these mode overlap plots: it is clear that lines from x and y-axes lie very close to each other. The important thing here is that at the "estimated" ITMX non-thermal lens value +15uD and the as-designed PR2-PR3 length, the IMC-PRX overlap is 99.5%(x) and 98.5%(y), and the PRX-XARM overlap is 99.7%(x) and 99.2%(y). If we allow for a +20uD (+50km) thermal lens on top of this value, these overlaps change to IMC-PRX 99.4%(x) 99.8%(y) and PRX-XARM 99.8%(x) 100%(y). In short: IMC-PRX and PRX-XARM should be mode matched well at cold and warm states if we believe the beam size measurements. Of course, the mode matching starts to suffer at a bit lower power than planned as a result of this, but PRX-XARM should still remain >97% for effectively all powers up to 125W. Also worth noting here is that for the mode matching, it doesn't matter right now whether it's the PR2-PR3 length that is off or the ITM non-thermal lens power, since the slope of the mode overlaps with respect to PR2-PR3 offset matches the diagonal lines from the beam size measurements.

I've also been looking at PRX-PRY overlaps for comparison with PRC gain observations as a function of ring heater power, but this post is already too long so I'll post that later.

* PR2-PR3 was varied in the simulation, but for these purposes it's degenerate with PR3 Rc change. The actual quantity that seems to matter most is (2 x PR2-PR3 distance - PR3 Rc), but in the simulation it suffices to vary just one of these. In practice we can only change PR2-PR3 distance so I plot that one.

**One caveat there: since the x-axis PRM measurement is believed to have been affected by the reflection from the back surface of the pick off window, I used the the model value of PRM direct reflected beam size and the measured value of the ITMX direct reflected beam size for computing the ratio. For comparison, the y-axis PRM direct reflected beam size (unaffected by the second reflection form the window) was measured to be 2.15mm, compared to the model value of 2.138mm. This is less than 1% difference, and the PRM model x-axis beam size was scaled by 2.15/2.138 to account for the possible small error in measurement location.