We tried to simulate the effects of differential ITM thermal lens on SRM sensing. The idea of differential ITM thermal lensing is motivated by alog 34853.
In the attachments we considered 4 different SRC geometries corresponding to G={0.76, 0.80, 0.83, 0.86} or one-way gouy phase={20.1, 18.6, 17.0, 15.3}, respectively. Note that here G=(A+D)/2, which is different from the stability g=arccos(gouy phase) = sqrt((A+D+2)/4).
*In each plot, the top panel showed the recycling gain relative to nominal (we defined the gain as the ratio of E_{-9}^ast E_{9} inside/outside PRC).
*The left two panels showed the AS36 signal as function of AS port gouy phase, with the upper panel showing the abs of AS36 to SRM pitch response, and the lower panel the difference of SRM and BS demodulation phase. The orange trace is for the nominal setting (based on the finesse input file T1300904), and the blue trace when differential ITM lensing introduced.
*The right two panel show an anatomy of the AS36 signal, i.e. the beat note of <9_00, 45_01>, <9_02, 45_01>, <9_01, 45_00>, <9_01, 45_02>. For conciseness we do not show the contribution from the negative sidebands.
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Some thing might be interesting to note:
1. When the ifo is at its nominal setting, the dominating AS36 signal comes from the beat between <9_00, 45_01>. This pair stays roughly constant w.r.t. different ITM diff lensing. However, because 9_00 is weak, when we add differential lensing, both <9_02, 45_01> and <9_01, 45_00> can be as large as, or even larger than <9_00, 45_01>, making the AS36 signal very messy.
2. If SRC gouy phase >~ 18 deg, if we happen to put the two WFS at AS gouy phase around 35 deg and 125 deg, we would basically have no srm signal at all. If this is indeed the cause, increasing SR2-SR3 distance alone might not be sufficient to solve the sensing problem.
3. If SRC gouy phase <~ 18 deg, the SRM signal becomes stronger as we adding diff lensing. However, it also becomes more degenerate with BS signal whose response is ~ 100 greater than SRM. If non of the WFS are placed at AS gouy phase [75, 140] deg, the SRM signal, despite higher amplitude, will also be lost due to degenerate demod phase with BS.
4. The diff lensing can reduce recycling gain. For ~75 km extra lens added to ITMX, the recycling gain decreases by ~5%.
5. Bad centering loops might not be necessary in explaining the bad srm sensing. As argued in T0810007, the carrier doesn't see ITM RoC mismatch to 1st order whereas the sidebands do. The effects due to carrier junk might not be as significant as sideband homs.
Thank you, Hang, for doing this. A few questions/suggestions.
- Have you looked at the carrier recycling gain as well?
- I am asking this because the H1 input file assumes ultra low loss on test masses.
- What is the definition of the angle shown in the anatomy panel?
- Why does the demod phase difference vary as a function of the Gouy phase? I naively thought that the demod and Gouy phases were independent...
Thanks Kiwamu for the comments.
* According to the simulation, the carrier recycling gain (which we approximated by the total DC power in PRC/ total DC before entering the ifo) does not vary significantly w/ extra ITM thermal lensing, which might be a consequence that the carrier doesn't see RoC mismatch to first order? The simulation is attached. The recycling gain doesn't care that much about the SRC so I just used the nominal 20 deg gouy SRC for plotting.
And you are right, the Finesse input file assumes an extremely loss ifo; for RF9 the build up is essentially the same as the build up inside an ideal FP cav w/ input T=0.03 and output T=0. We haven't touched the losses for now, however, as adding loss somewhere to reduce the recycling gain can be a very degenerate process. But as suggested in Aidan's log the extra absorption is very localized, meaning a thermal lens should not capture all the physical effects. Our next step will be adding some extra ITMX loss to our parameter searching space and explore what effect that may cause.
* For angles in the anatomy panel, we defined (w/ abused notation) <f1_{TEM nm}, f2_{TEM n'm'}> = (geometrical pre-factor) E*(f1)_{nm} ·E(f2)_{n'm'}, where E represent the complex-valued field. Therefore the beatnote at (f2-f1) is also complex. And the angle of this complex value is what we plotted. Or in other words, this angle is the phase difference of the two fields that beat to 36.
* We defined the AS response as AS_I + 1j * AS_Q, and we called the demodulation phase as the phase angle of this complex quantity. Or more specifically it is the phase at a given gouy phase that maximizes the signal in the I-quadrature. Because AS_I and AS_Q sees the AS gouy phase differently, angle(AS_I + 1j*AS_Q) is thus gouy phase dependent.
