EDIT: some images look bad on the elog, and the notebook is parsed, which is is bad. Almost everything posted here is in the compressed file attachment.
As we've been discussing, we want to reduce the laser's jitter effect on the QPDs of the OpLevs, without losing sensitivity to angular motion of the mirror; the current setup is roughly described in this picture:
The idea is to place an additional lens (or lenses) between the mirror and the QPD, as shown in the proposed setup in this picture:
I did some ray tracing calculations to find out how the system would change with the addition of the lens. The step-by-step calculations are done at the several points shown in the pictures, but here I will just summarize. I chose to put the telescope at a variable relative distance x from the QPD, such that x=0 at the QPD, and x=1 at the mirror.
Here are the components that I used in the calculations:
I used a 3x3 matrix formalism in order to have easier calculations and reduce everything to matrix multiplications; that because the tilted mirror has an annoying addictive term, which I could get rid of:
Therefore, n the results the third line is a dummy line and has no meaning.
For the first case (first schematic), we have, for the final r and Theta seen at the QPD:
In the second case, we have a quite heavy output, which depend also on x and f:
Now, some plots to help understand the situation.
What we want if to reduce the angular effect on the laser displacement, without sacrificing the sensitivity on the mirror signal. I defined two quantities:
Beta is the laser jitter we want to reduce, while Gamma is the mirror signal we don't want to lose. I plotted both of them as a function of the position x of the new lens, for a range of focal lengths f. I used d1 = d2 = 2m, which should be a realistic value for the 40m's OpLevs.
Plot of Beta
Plot of Gamma
Even if it is a bit cluttered, it is useful to see both of the same plot:
Plot of Beta & Gamma
Apart from any kind of horrific mistakes that I may have done in my calculations, it seems that for converging lenses our signal Gamma is always reduced more than the jitter we want to suppress. For diverging lenses, the opposite happens, but we would have to put the lens very near to the mirror, which is somehow not what I would expect. Negative values of Beta and Gamma should mean that the final values at the QPD level are on the opposite side of the axis/center of symmetry of the QPD with respect to their initial position.
I will stare at the plots and calculations a bit more, and try to figure out if I missed something obvious. The Mathematica notebook is attached.