Zernike Polynomials

The circle polynomials of Zernike, or Zernike polynomials, are an infinite set of polynomials orthonormal on the unit circle. They are described in detail by, for example, Born and Wolf in their well-known “Principles of Optics” book. We use these polynomials as the basis set to parametrise the wavefront errors present in a telescope because:

  • Low order polynomials correspond to large scale wavefront errors which are well constrained by the OOF technique.
  • Some of the polynomials correspond to well known aberrations which are often present in telescopes.

C++ Code for generating Zernike polynomials

The code used by the OOF system to calculate Zernike polynomials is available under the terms of the GPL. If you are interested, from the software page download the bnlib component and look for the file named “zernikepoly.hxx/cxx”.

Plots of the polynomials

The table below shows the first twenty Zernike polynomials and the corresponding model beams. The images are actually 512x512 resolution, and you can view them at this resolution be either saving them to disk or, in Firefox or Mozilla, right-clicking on the image and selecting View Image.

The first column in the table shows some of the labels that are used for the Zernike polynomial shown. The n=,l= label shows the radial (n) and angular (l) order of the polynomial. The OOF= label shows the label used by the OOF software. The GBT= label is the label used by the GBT control software.

The second column shows a representation of wavefront phase corresponding to the Zernike polynomial. It has been calculated on an unit circle and the amplitude of the polynomial is one radian of phase at the circle (or aperture) edge.

The third, fourth and fifth columns show the calculated in-focus, +ve out of focs and -ve out of focus beams corresponding to an aperture with aberrations shown in the second columns. The magnitude of the de-focus is two radians and the edge of the aperture.

Plots of Zernike polynomials typically used in the OOF holography fitting and of corresponding in-focus and out-of-focus beams:

Label Phase In-focus beam +ve out-of-focus beam -ve out-of-focus beam
n=1, l=-1, OOF=1, GBT=3 z-1-1 z-1-1-dz0 z-1-1-dz2 z-1-1-dz-2
n=1, l=1, OOF=2, GBT=2 z-11 z-11-dz0 z-11-dz2 z-11-dz-2
n=2, l=-2, OOF=3, GBT=6 z-2-2 z-2-2-dz0 z-2-2-dz2 z-2-2-dz-2
n=2, l=0, OOF=4, GBT=5 z-20 z-20-dz0 z-20-dz2 z-20-dz-2
n=2, l=2, OOF=5, GBT=4 z-22 z-22-dz0 z-22-dz2 z-22-dz-2
n=3, l=-3, OOF=6, GBT=1 z-3-3 z-3-3-dz0 z-3-3-dz2 z-3-3-dz-2
n=3, l=-1, OOF=7, GBT=9 z-3-1 z-3-1-dz0 z-3-1-dz2 z-3-1-dz-2
n=3, l=1, OOF=8, GBT=8 z-31 z-31-dz0 z-31-dz2 z-31-dz-2
n=3, l=3, OOF=9, GBT=7 z-33 z-33-dz0 z-33-dz2 z-33-dz-2
n=4, l=-4, OOF=10, GBT=15 z-4-4 z-4-4-dz0 z-4-4-dz2 z-4-4-dz-2
n=4, l=-2, OOF=11, GBT=14 z-4-2 z-4-2-dz0 z-4-2-dz2 z-4-2-dz-2
n=4, l=0, OOF=12, GBT=13 z-40 z-40-dz0 z-40-dz2 z-40-dz-2
n=4, l=2, OOF=13, GBT=12 z-42 z-42-dz0 z-42-dz2 z-42-dz-2
n=4, l=4, OOF=14, GBT=11 z-44 z-44-dz0 z-44-dz2 z-44-dz-2
n=5, l=-5, OOF=15, GBT=21 z-5-5 z-5-5-dz0 z-5-5-dz2 z-5-5-dz-2
n=5, l=-3, OOF=16, GBT=20 z-5-3 z-5-3-dz0 z-5-3-dz2 z-5-3-dz-2
n=5, l=-1, OOF=17, GBT=19 z-5-1 z-5-1-dz0 z-5-1-dz2 z-5-1-dz-2
n=5, l=1, OOF=18, GBT=18 z-51 z-51-dz0 z-51-dz2 z-51-dz-2
n=5, l=3, OOF=19, GBT=17 z-53 z-53-dz0 z-53-dz2 z-53-dz-2
n=5, l=5, OOF=20, GBT=16 z-55 z-55-dz0 z-55-dz2 z-55-dz-2

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