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Rotationally Invariant Polynomials:
These polynomials must have m 0 .
The set of rotationally invariant polynomials that exist in the current set of 36 Zernike
terms described in ISO 10110-5, Appendix A.3, first edition are as follows:
Z
2,0
is
Z
3
r , 2r
2
1
Z
4,0
is
Z
8
r , 6r
4
6r
2
1
Z
6,0
is
Z
15
20r
6
30r
4
12r
2
1
Z
8,0
is
Z
24
70r
8
140r
6
90r
4
20r
2
1
Z
10,0
is Z
35
r , 252r
10
630r
8
560r
6
210r
4
30r
2
1
These polynomials have no azimuthal dependence and describe the wavefront aspheric
approximation f
wri
in section 3.2.6. Additional rotationally invariant terms can be
added to more accurately approximate the wavefront.
Rotationally Varying Polynomials:
These polynomials must have m 0 .
Rotationally varying polynomials have a radial function as well as a sine or cosine
function which dictates the order of the azimuthally varying function.
These polynomials come in pairs, having an equal value for n and the positive/negative
equal integer walue for m.
For example:
Z
2,2
is astigmatism about 0 degrees.
r
2
cos
Z
2, 2
is astigmatism about 45 degrees.
r
2
sin
A vector summation along with the arc tangent expression will give the magnitude and
the directio of the resultant pair of matching polynomials.
Directional Angle 1 m arctan Z
n , m
Z
n ,m
Magnitude 2
Z
n , m
2
Z
n , m
2
The Zernike Polynomial Term is written:
Z
n , m
where:
n = radial order
m = azimuthal order
n is an integer > 0.
n - |m| must be a non-negative even integer including 0.
If
n = 8; then m can equal 8, 6, 4, 2, -2, -4, -6, -8 only.
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Definitions:
Z
n , m
is the coefficient value for the expression:
R
nm
r
sin m
for m is a negative number
R
nm
r
cos m
for m is a positive number
And further: R
nm
r is the radial function:
R
nm
r
s 0
l
1
s
n s
s l s n l s r
n 2s
where l n m
2
sin m a and cos m a are the azimuthal functions
Example:
Z
8, 4
(pronounced, Z eight minus four) would be the magnitude of the Zernike
coefficient for the following equation.
28r
8
42r
6
15r
4
sin 4
Which is an 8
th
order radial function and 4
th
order azimuthal function.