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* Still need to finish calculating what could be causing our big arm power fluctuations (Test mass angular motion? PRM angular motion? ALS noise?) (Calculation)

I think that our problem of seeing significant arm power fluctuations while we bring the arms into resonance during PRMI+arms tests is coming from PRM motion. I've done 3 calculations, so I will describe below why I think the first two are not the culprit, and then why I think the PRM motion is our dominant problem.
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ALS length fluctuations
Arm length fluctuations seem not to be a huge problem for us right now, in terms of what is causing our arm power fluctuations.
What I have done is to calculate the derivative of the power in the arm cavity, using the power buildup that optickle gives me. The interferometer configuration I'm using is PRFPMI, and I'm doing a CARM sweep. Then, I look at the power in one arm cavity. The derivative gives me Watts buildup per meter CARM motion, at various CARM offsets. Then, I multiply the derivative by 60 nm, which is my memory of the latest good rms motion of the ALS system here at the 40m. I finally divide by the carrier buildup in the arm at each offset, to give me an approximation of the RIN at any CARM offset.
I don't know exactly what the calibration is for our ALS offset counts, but since we are not seeing maximum arm cavity buildup yet, we aren't very close to zero CARM offset.
From this plot, I conclude that we have to be quite close to zero offset for arm length fluctuations to explain the large arm power fluctuations we have been seeing.
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AS port contrast defect from ETM motion
For this calculation, I considered how much AS port contrast defect we might expect to see given some ETM motion. From that, I considered what the effect would be on the power recycling buildup.
Rather than doing the integrals out, I ended up doing a numerical analysis. I created 2 Gaussian beams, subtracted the fields, then calculated the total power left. I did this for several separations of the beams to get a plot of contrast defect vs. separation. My simulated Gaussian beams have a FWHM of 1 unit, so the xaxis of the plot below is in units of spot motion normalized by spot size.
Unfortunately, my normalization isn't perfect, so 2 perfectly constructively interfering beams have a total power of 0.3, so my yaxis should all be divided by 0.3.
The actual beam separation that we might expect at the AS port from some ETM motion (of order 1e6 radians) causing some beam axis shift is of the order 1e5 meters, while the beam spot size is of the order 1e3 meters. So, in normalized units, that's about 1e2. I probably should change the xaxis to log as well, but you can see that the contrast defect for that size beam separation is very small. To make a significant difference in the power recycling cavity gain, the contrast defect, which is the Michelson transmission, should be close to the transmission of the PRM. Since that's not true, I conclude that ETM angular motion leading to PRC losses is not an issue.
I still haven't calculated the effect of ITM motion, nor have I calculated either test mass' angular effect directly on arm cavity power loss, so those are yet to be done, although I suspect that they aren't our problem either.
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PRM motion
I think that the PRM moving around, thus causing a loss in recycling gain, is our major problem.
First, how do I conclude that, then some thoughts on why the PRM is moving at all.
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theta = 12e6 radians (ref: oplev plot from elog 9338 last week)
L = 6.781 meters
g = 0.94
a = theta * L /(1g) = 0.0014 meters axis displacement
w0 = 3e3 meters = spot size at ITM
a^2/w0^2 = 0.204 ==>> 20% power loss into higher order modes due to PRM motion.
That means 20% less power circulating, hitting the ITMs, so less power going into the arm cavities, so less power buildup. This isn't 50%, but it is fairly substantial, using angular fluctuation numbers that we saw during our PRMI+arms test last week. If you look at the oplev plot from that test, you will notice that when the arm power is high (as is POP), the PRM moves significantly more than when the carrier buildup in the cavities was low. The rms motion is not 12 urad, but the peaktopeak motion can occasionally be that large.
So, why is that? Rana and I had a look, and it is clear that there is a difference in PRM motion when the IFO is aligned and flashing, versus aligned, but PSL shutter is closed. Written the cavities flash, the PRM gets a kick. Our current theory is that some scattered light in the PRC or the BS chamber is getting into the PRM's OSEMs, causing a spike in their error signal, and this causes the damping loops to push on the optic.
We should think a little more on why the PRM is moving so much more that any other optic while the power is building up, and if there is anything we can do about the situation without venting. If we have to, we should consider putting aluminum foil beam blocks to protect the PRM's OSEMs. 