The ef wavefront reduction package. -- 1.2 Zernike representation

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1.2 Zernike representation

By default the Zernikes within the program are normalized such that each has unit RMS wavefront. This normalization, and the default Zernike ordering is according to Noll . This normalisation may be converted to peak normalsation by multiplying by sqrt(n+1) when m=0 or sqrt(2*(n+1)) when m != 0.

Noll Number Zernike names Conversion to peak normalisation

2,3 Tip and tilt sqrt(4) = 2

4 Focus sqrt(3) = 1.73

5,6 Astigmatism sqrt(6) = 2.45

7,8 Coma sqrt(8) = 2.83

9,10 Trefoil sqrt(8) = 2.83

11 Spherical sqrt(5) = 2.23

12,13 Quad astigmatism sqrt(10) = 3.16

14,15 r⁴ exp4i sqrt(10) = 3.16

16,17 r⁵ expi sqrt(12) = 3.46

18,19 r⁵ exp3i sqrt(12) = 3.46

20,21 r⁵ exp5i sqrt(12) = 3.46

22 Spherical sqrt(7) = 2.65

Converting Zernike coefficients from RMS normalisation to peak normalsation.

All internal calculations are carried out with full disk Zernikes, which means that care must be taken when reducing data from optical systems with large central obscurations. In general with obscuration ratios greater then 33%, do not try to correct the extra-focal images for Zernikes higher than 13. For obscurations less than 33%, the use of full disk Zernikes does not affect the accuracy of the results.

Laplacian Optics Inc. Email: laplace@laplacian.com

Noll Number	Zernike names	Conversion to peak normalisation
2,3	Tip and tilt	sqrt(4) = 2
4	Focus	sqrt(3) = 1.73
5,6	Astigmatism	sqrt(6) = 2.45
7,8	Coma	sqrt(8) = 2.83
9,10	Trefoil	sqrt(8) = 2.83
11	Spherical	sqrt(5) = 2.23
12,13	Quad astigmatism	sqrt(10) = 3.16
14,15	r⁴ exp4i	sqrt(10) = 3.16
16,17	r⁵ expi	sqrt(12) = 3.46
18,19	r⁵ exp3i	sqrt(12) = 3.46
20,21	r⁵ exp5i	sqrt(12) = 3.46
22	Spherical	sqrt(7) = 2.65