Zernike PolynomialsΒΆ
The Zernike polynomials are a complete sequence of polynomials that are orthogonal on the unit disk. Using polar coordinates
, so that
, the Zernike polynomials are defined as
with
and where the integer index pair is given by
where , and
represents the largest integer that is less or equal to the delimited integer.
Warning
Different orderings of the Zernike polynomials are in use. Here, we followed
[1] (page 213). Besides this, different scalings of the Zernike polynomials are used.
In the above, the Fringe convention as been used for scaling (c.f. http://en.wikipedia.org/wiki/Zernike_polynomials or http://mathworld.wolfram.com/ZernikePolynomial.html). For the sake of clarity, the following table lists the leading 36 (Fringe)-Zernike polynomials:
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description |
---|---|---|---|---|
1 | 0 | 0 | ![]() |
piston |
2 | 1 | 1 | ![]() |
x-tilt |
3 | 1 | -1 | ![]() |
y-tilt |
4 | 2 | 0 | ![]() |
defocus |
5 | 2 | 2 | ![]() |
astigmatism |
6 | 2 | -2 | ![]() |
astigmatism |
7 | 3 | 1 | ![]() |
coma |
8 | 3 | -1 | ![]() |
coma |
9 | 4 | 0 | ![]() |
spherical aberration |
10 | 3 | 3 | ![]() |
trifoil |
11 | 3 | -3 | ![]() |
trifoil |
12 | 4 | 2 | ![]() |
astigmatism |
13 | 4 | -2 | ![]() |
astigmatism |
14 | 5 | 1 | ![]() |
coma |
15 | 5 | -1 | ![]() |
coma |
16 | 6 | 0 | ![]() |
spherical aberration |
17 | 4 | 4 | ![]() |
four wave |
18 | 4 | -4 | ![]() |
four wave |
19 | 5 | 3 | ![]() |
trifoil |
20 | 5 | -3 | ![]() |
trifoil |
21 | 6 | 2 | ![]() |
astigmatism |
22 | 6 | -2 | ![]() |
astigmatism |
23 | 7 | 1 | ![]() |
coma |
24 | 7 | -1 | ![]() |
coma |
25 | 8 | 0 | ![]() |
spherical aberration |
26 | 5 | 5 | ![]() |
five wave |
27 | 5 | -5 | ![]() |
five wave |
28 | 6 | 4 | ![]() |
four wave |
29 | 6 | -4 | ![]() |
four wave |
30 | 7 | 3 | ![]() |
trifoil |
31 | 7 | -3 | ![]() |
trifoil |
32 | 8 | 2 | ![]() |
astigmatism |
33 | 8 | -2 | ![]() |
astigmatism |
34 | 9 | 1 | ![]() |
coma |
35 | 9 | -1 | ![]() |
coma |
36 | 10 | 0 | ![]() |
spherical aberration |
Bibliography
[1] | Gross H. (editor), Handbook of Optical Systems, Volume III, Wiley-VCH 2005 |