THORLABS WFS20-14AR, Operation Manual

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© 2007 - 2018 Thorlabs
8 Appendix
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8.5 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 their amplitudes or Zernike coefficients c
n
. Generally a least square Zernike fit is
done in order to determine the Zernike coefficients
c
n
.
A variety of definitions exist for these Zernike functions. The Thorlabs Wavefront Sensor soft-
ware and this manual follows the
ANSI Standard Z80.28-210 , see literature [1] .
The Zernike polynomials are recommended for describing wave aberration functions over circu-
lar pupils with a unit radius. Individual terms or modes of a Zernike polynomial are mutually or-
thogonal 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.
Rectangular coordinates are x any y both normalized to unit radius 1.
Polar coordinates (
r
,
q
) are normalized radius coordinate
r
and angular coordinate
q,
where
r
=
r/a
with
r
as physical radial coordinate and
a
as the pupil radius.
The natural scheme for ordering of the Zernike modes is to use a double index corresponding
to the radial order
n and angular frequency m as following equation.
n
Radial Order
| m n Angular frequency
r
normalized radial coordinate
q
(0 .. 2 π ) angular coordinate
r physical radial coordinate
a
physical pupil radius
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.
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