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Message ID: 372     Entry time: Fri Aug 23 11:11:44 2019
 Author: shruti Type: Optics Category: Characterization Subject: Finding the curvature bottom

I attempted to fit the data taken by Koji of the beam spot precession at the CCD in order to find the location of the curvature bottom in terms of its distance (d) and angle ($\phi$) from the centre of the mirror. This was done using the method described in a previous similar measurement  and Section 2.1.3 of T1500060.

Initially, I attempted doing a circle_fit on python as seen in Attachment 1, and even though more points seem to coincide with the circle, Koji pointed out that the more appropriate way of doing it would be to fit the following function:

$f(i, \theta, r, \phi) = \delta_{i,0} [r \cos(\theta+\phi) + x_c] + \delta_{i,1} [r \sin(\theta+\phi) +y_c]$

since that would allow us to measure the angle $\phi$ more accurately; $\phi$ is the anti-clockwise measured angle that the curvature bottom makes with the positive x direction.

As seen on the face of the CCD, x is positive up and y is positive right, thus, plotting it as the reflection (ref. Attachment 2) would make sure that $\phi$ is measured anti-clockwise from the positive x direction.

The distance from the curvature bottom is calculated as

$d = \frac{rR}{2L}$

r: radius of precession on CCD screen (value obtained from fit parameters, uncertainty in this taken from the std dev provided by fit function)

R: radius of curvature of the mirror

L: Distance between mirror and CCD

R = 2.575 $\pm$ 0.005 m (taken from testing procedure doc referenced earlier) and L = 0.644 $\pm$ 0.005 m (value taken from testing doc, uncertainty from Koji)

d (mm) $\phi$ (deg)
C7 0.554 $\pm$ 0.004 -80.028 $\pm$ 0.005
C10 0.257 $\pm$ 0.002 -135.55 $\pm$ 0.02
C13 0.161 $\pm$ 0.001 -79.31 $\pm$ 0.06

 Attachment 1: CircleFit.pdf  81 kB  Uploaded Fri Aug 23 13:26:45 2019
 Attachment 2: SineFit.pdf  159 kB  Uploaded Fri Aug 23 13:28:57 2019
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