THORLABS WFS150, Operation Manual

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109 Appendix
© <2007> ... Thorlabs GmbH
7 Appendix
7.1 Zernike Fit and Zernike Modes
The wavefront can be expressed in terms of a set of Zernike functions, also
called Zernike modes.
The wavefront W(x,y) is derived from a summation of orthonormal Zernike
functions Z
n
(x,y) weighted by its amplitudes or Zernike coefficients c
n
. Generally a
least square Zernike fit is done in order to determine the Zernike coefficients c
n
.
There exist various definitions about these Zernike functions. The Thorlabs
Wavefront Sensor software and this manual follows the convention according to
Malacara , see literature.
The Zernike polynomials are recommended for describing wave aberration
functions over circular pupils with unit radius. Individual terms or modes of a
Zernike polynomial are mutually orthogonal over the unit circle and are easily
normalized to form an orthonormal basis. Zernike modes are conveniently
expressed in either
polar or rectangular coordinate reference frames.
Polar coordinates (r, q ) are radius r and radial angle q, whereas r = r /a with r as
polar radial dimension and a as the pupil radius. Rectangular coordinates are x
any y both normalized to unit radius 1.
The natural scheme for ordering of the Zernike modes is to use a double index
corresponding to the
radial order and angular frequency as shown above in the
above equation. However, for numerical purposes it is useful to have a single-
index scheme which allows the Zernike coefficients to be written as a vector for
use in linear algebra calculations. Numerous schemes for ordering the Zernike
modes have been proposed in the optics literature, each with its own advantages
and disadvantages.
The following table gives the Zernike polynomials in radial coordinates of order 0 -
4 according to the Malacara index scheme.