I stared a bit longer at the plots and thanks to Eric's feedback I noticed I payed too much attention to the comparison between Beta and Gamma and not enough attention to the fact that Beta has some zero-crossings...
I made new plots, focusing on this fact and using some real values for the focal lengths; some of them are still a bit extreme, but I wanted to plot also the zero-crossings for high values of `x`, to see if they make sense.
### Plot of Beta and Gamma
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### Plot of Beta and Gamma (zoom)
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If we are not interested in the sign of our signals/noises (apart from knowing what it is), it is maybe more clear to see regions of interest by plotting Beta and Gamma in absolute value:
### Plot of Beta and Gamma (Abs)
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I don't know if putting the telescope far from the QPD and near the mirror has some disadvantage, but that is the region with the most benefit, according to these plots.
The plots shown so far only consider the coefficients of the various terms; this makes sense if we want to exploit the zero-crossing of Beta's coefficient and see how things work, but the real noise and signal values also depend on the Alpha and Theta themselves. Therefore I made another kind of plot, where I put the ratio r'(Alpha)/r'(Theta) and called it Tau. This may be, in a very rough way, an estimate of our "S/N" ratio, as Alpha is the tilt of the mirror and Theta is the laser jitter; in order to plot this quantity, I had to introduce the laser parameters r and Theta (taken from the Edmund Optics 1103P datasheet), and also estimate a mean value for Alpha; I used Alpha = 200 urad. In these plots, the contribute of r'(r) is not considered because it doesn't change adding the telescope, and it is overall small.
In these plots the dashed line is the No Telescope case (as there is no variable quantity), and after the general plot I made two zoomed subplots for positive and negative focal lengths.
### Plot of Tau (may be an estimate of S/N)
### Plot of Tau (positive f)
### Plot of Tau (negative f)
If these plot can be trusted as meaningful, they show that for negative focal lengths our tentative "S/N" ratio is always decreasing which, given the plots shown before, it does little sense: although for these negative f Gamma never crosses zero, Beta surely does, so I would expect one singular value each. |