© 2007-2012 Thorlabs GmbH
137Appendix
6.3.8
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, ) are radius r and radial angle whereas r = /a with 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.
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