# 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:

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 |