# Donut Analysis Details¶

## Aberration to Hexapod Transfer Matrix¶

The Donut analysis ultimately measures 4 quantities used to derive the alignment. These are the AstigmatismY(ThetaX), AstigmatismX(ThetaX), Delta ComaY, Delta ComaX (Zern5 = AstigY, Zern6=AstigX, Zern7=ComaY, Zern8=ComaX). A set of images were taken with large decenters and tip/tilts, here are shown plots of the four Aberrations vs. Hexapod degrees of freedom. Note that the scales in the plots are not all the same, 8 of the plots have very nice fits and big slopes, while the other 8 have only small variations and poor fits. The latter 8 are close to the expectation from Zemax - ie. zero slope. I use 0.0 as the slope for these 8, although revisiting this, two of these offdiagonal terms are clearly small but non-zero and there would be some improvement from including these terms. Then I take the slopes of these 4x4 plots, invert their matrix, and use that as the transfer matrix.

Thus the slope matrix d(Aberration)/d(Hexapod) is:

[[ 0.00 , 4.25e-07, -2.18e-05, 0.00 ],
[ 3.87e-07, 0.00 , 0.00 , 2.17e-05],
[ 2.11e-04, 0.00 , 0.00 , -5.09e-04],
[ 0.00 , -2.26e-04, -6.10e-04, 0.00 ]]

and inverting gives the transfer matrix d(Hexapod)/d(Aberration) :

[[ 0.00e+00 1.07e+05 4.54e+03 0.00e+00]
[ 1.18e+05 -0.00e+00 0.00e+00 -4.20e+03]
[ -4.36e+04 0.00e+00 0.00e+00 -8.20e+01]
[ 0.00e+00 4.42e+04 -8.10e+01 0.00e+00]]

Here is the matrix found by doing the same thing in Zemax - the terms compare to within 30%-70% of the simulated values.

[[ 0.00e+00 1.72e+05 3.44e+03 0.00e+00]
[ 1.72e+05 0.00e+00 0.00e+00 -3.44e+03]
[ -3.87e+04 0.00e+00 0.00e+00 -5.88e+01]
[ 0.00e+00 3.87e+04 -5.88e+01 0.00e+00]]