# Zernike Polynomials¶

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:

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