Summary: in preparation for beam arriving in HAM6 / ISCT6, I wanted to explore the potential for mode mismatch to the OMC, and how it might be corrected.  I found that for essentially any reasonable combination of errors in optic positions and ROCs, the mode matching can be recovered by a small adjustment to SR2.  (Given the way the SRC is designed, I don't think this is surprising to anyone -- it may even be intentional! -- but it was an interesting exercise.)

Details:

At L1 they observe a mode mismatch to the OMC of ~25%, depending on which ITM supplies the bounce.  Lisa found that this could be caused by a small (1.5cm) change to the SR2-SR3 distance, assuming the PRC length is nominal.  At LHO we are a 2-3 weeks away from measuring the beam arriving in HAM6.  In principle, there may be errors in the position of any of the optics on the order of 1cm, and errors to the ROC for the curved mirrors (of order ~few cm?).  The question is, if we are very unlucky and the initial mode matching to the OMC is bad, can we correct it in a simple way?

I used Lisa's script from LLO:8565 as a starting point to estimate the mode mismatch that could occur from small errors in the positions and ROCs for optics in the output path, SR3 to OMC.  At LHO the PRC length has been measured to better than 1mm; it's very close to nominal, so for now I assume the positions of the PRs, the BS, and the ITMs are correct.  That leaves six optical components with un-verified positions: SR3, SR2, SRM, OM1, OM2, and the OMC.  (I fold errors in the position of OM3 into the position of the OMC.)

Modeling a beam subject to small variations in eleven optical parameters is a lot to keep track of in closed-form, so I implemented a Monte Carlo approach: for 10k trials I independently varied the longitudinal position and ROC of the six optics in the output path, and calculated the mode overlap with the OMC waist (w0=490um).

Errors in position were drawn from Gaussian random variables with sigma = 2.0cm; these were applied to SR3, SR2, SRM, OM1, OM2, and the OMC.  Errors in radius of curvature were drawn from a Gaussian distribution with sigma = 5.0cm; these were applied to SR3, SR2, SRM, OM1, and OM2.  I'm not sure if these values are reasonable (2cm in position sounds like a lot), but they seemed like fair conservative guesses, based on the as-built dimensions for L1 in E1200274-v3, compared to the nominal values in T0900043-v11.

For 10k trials, the median overlap with the OMC waist after varying the parameters of the optics was 0.85; the distribution is shown in Fig2.  This median is better than what's observed at L1, which may mean they got unlucky, or the magnitudes of my errors are too small.  (NOTE: for simplicity I am using a single bounce off ITMX with the nominal ROC of 1934m.)

Next, I used a la mode's optimizePath() function to correct the mode mismatch by varying the position of SR2.  Based on table layouts this seemed to be the easiest optic to move.  The range on the optimization of SR2's position was +/-5cm.

The result is that even for very bad mode overlaps, the errors can be compensated by moving SR2.  And, whether or not the overlap can be completely recovered is only a function of how far you can move SR2.  (I.e., if we are terribly unlucky at H1, maybe we can move it by more than 5cm.)  This might be known already to optics experts, but it was surprising to me that even for large errors in optic ROCs the mode can be corrected by changing a single degree of freedom.  I guess this is what you gain when your beam-reducing telescope has a short Rayleigh range?  (Flip side: we're really sensitive to the position of SR2 and SR3.)

In the attached: Fig1 is the distribution of mode overlap to the OMC, for 10k trials with independently varied parameters.  Fig2 is how well you can improve things by moving SR2; the horizontal coordinate is starting (mis)match, and the vertical coordinate is corrected (mis)match, after at most a +/-5cm change to SR2.  Fig3 is a comparison of how much you need to move SR2 vs how much you get back.  The scripts I used are there too.  It's not a very elegant implementation, for 10k trials it takes way too long to finish, something like an hour.



Notes:

 - This is all fine from a mode-matching perspective, but I don't know enough about optical cavities to say whether changing the SRC length by 5cm is okay or a complete disaster.  Also, I think that a la mode's optimization procedure changes only the position of the optic in question, and doesn't take into account the changes to relative lengths.  So, when it moves SR2 by 5cm, a la mode is increasing the distance from SR3 to SR2, and decreasing the distance from SR2 to SRM.  This would be fine if SR2 was a lens, but it's a mirror; if the position changes by 5cm the SR3-SR2 and SR2-SRM distances should change in the same direction.  (I think that since the SR3-SR2 distance is the important one, this is does not change the results, but I haven't checked in detail.)

  - I belatedly realized that the ROCs for the SR optics have been measured and they're listed on the core optics website (galaxy.ligo.caltech.edu/optics).  So, errors in the ROC of 5cm are probably way too generous.  I'm not sure about the OM1 and OM2 optics.

 - Of course in order to correct something you need to measure it first.  If the mode mismatch is bad we'll have to characterize the beam on ISCT6 with Chris M's beam scan technique or something similar.  It might be worth modeling how accurately we can measure the necessary correction to SR2's position.