20130412 Horizontal Fitting

Horizontal laser-wire scans were often non-Gaussian and although it was known that they were not strictly Gaussian, they were in the past close enough that a Gaussian approximation worked and was only a few percent wrong in the fitted sigma. The obvious solution is to use the overlap integral, but for this, the vertical size needs to be known - catch 22. However, there is some solice in the knowledge that the vertical size is likely to be between 0.5 and 1.5 microns.

Using the horizontal scans from the quad scans, the overlap integral was used to fit each scan for a range of different vertical sigmas to see the effect this would have on the fitted horizontal sigma. This is shown in the figure below.

and zoomed in a little...

This shows that the likely uncertainty on the horizontal size given the knowledge of the vertical beam size is likely to be up to 11% (at the focus of the quad scan - where ~$\sigma_{x}$/x = 15/130 = 11% )

For these fits it's also possible to look at the reduced chi squared to see how well the model fits for these different parameters. This is shown in the figure below.

These show varying results, but generally a lower reduced chi squared between vertical sigma = 0.5 -> 1.5um for most cases than at other vertical sizes. Again, these show greatest sensitivity at the middle of the quad scan - the focus. Whilst you would normally expect the horizontal beam size to vary only slightly and generally in a linear way, with the specific optics that were in place during this scan, the (single) quad scan seem to focus in both the horizontal and the vertical! This is again shown in the figure below.

and zoomed into the focus...

Whatever the reason for this - perhaps strong coupling? - it is clear from the data.

Knock on Effect on Vertical

So this will have a clear effect on the measured vertical size. Below are the two graphs of the different horizontal fit models for the -92A case in the quad scan (approximately at the focus) on the last quad scan on 20130301. The horizontal data file is 20130301_1138_lws.dat and the nonlinear vertical scan is 20130301_1147_lws.dat

This shows a considerably better fit to the horizontal with the Overlap integral fit. For the first fit, a vertical sigma of 1.0um was assumed.

• sigma_ey = 1.0um -> sigma_ex = 135.66 +- 4.80 (overlap integral fit); reduced chi2 = 3.12
• sigma_ey = NA. -> sigma_ex = 215.75 +- 5.13 (Gaussian fit); reduced chi2 = 10.94

Propagating this to the horizontal, the vertical fits are:

• sigma_ex = 216um -> sigma_ey = 0.62 +- 0.07um; reduced chi2 = 17.91
• sigma_ex = 136um -> sigma_ey = 1.30 +- 0.05um; reduced chi2 = 4.34

The fit with the smaller horizontal sigma is a better fit both by eye and by the reduced chi2. The vertical size is consequently larger and is therefore different from the assumed vertical size in the horizontal fit. Putting this back into the horizontal fit yeilds:

• sigma_ey = 1.3um -> sigma_ex = 131.12 +- 4.91um; reduced chi2 = 3.13
• sigma_ex = 131.13um -> sigma_ey = 1.35 +- 0.05um; reduced chi2 = 4.1

Iterating again...

• sigma_ey = 1.35um -> sigma_ex = 130.34 +- 4.93um; reduced chi2 = 3.14
• sigma_ex = 130.34um -> sigma_ey = 1.36 +- 0.05um; reduced chi2 = 4.07

and again...

• sigma_ey = 1.36um -> sigma_ex = 130.19 +- 4.93um; reduced chi2 = 3.14
• sigma_ex = 130.19um -> sigma_ey = 1.36 +- 0.05um; reduced chi2 = 4.07

So no further improvement in reduced chi2 nor reduction in uncertainties (chi2 contour)

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