Zernike decomposition of holography beam

[JulienNGirard]

This is what I submitted to Kela, PhD student at Rhodes, working on beam modelling.

Zernike decomposition (orthogonal basis over a disk) of beams

Please see the sources:

• Zernike polynomials wikipedia

They are ordered using two parameters and they look like this:

• Python implementation found in PyCourse

First, by observing solely the basis vectors, I can tell that:

• Z₀⁰ & Z₂⁰ (and more generally Z_i⁰) can help fitting the first lobe and successive sidelobes.
• Z₃^-1 & Z₃¹ and Z₅^-1 & Z₅¹ can help fitting the pointing errors in the residuals (it ressembles what you had when you simulated pointing errors).
• Z₄^-2 & Z₄² can help fitting the particular 4-fold sidelobes of the VLA antenna.

To facilitate the access to this transform, I written/adapted it and uploaded a python notebook which make a decomposition of one slice of "shiftedscaled_ant5LLreal.fits" file available on jake. But can be updated to any holography beam.

Following are test plots to decompose the 1000th slice:

Decomposition using 100 Zernike polynomials in decomposition order

Same coefficients but in decreasing amplitude order, and selection of the 10 most powerful coefficients:

Comparisons of input data, "full" reconstruction (with 100 coeff), "truncated" reconstruction (with 10 coeff) along with error maps (2nd line). Each line is normalized to the same colormap.

As expected, the fewer coefficients, the larger the error compared to the input data to fit. Again, the Mean Square Error has to be evaluated properly.

Note about pretty plotting: At the end of the notebook you will find commands to edit pretty pictures (multiple plot, colormap normalized by line and a single corresponding colorbar at the end of the line). This is one way to improve and optimize your figures to put into a paper/thesis. Ok, my bad there are some typos on the plots I show.

[bennahugo]

THanks this is really cool stuff