Zernike Surface
Zernike Surface Information
KostaCLOUD offers various definitions to Zernike Surfaces, as every field, institution, and researcher has their preference to the ordering of Zernike Polynomials, normalization, etc. For this reason, KostaCLOUD allows you to pick your representation.
In KostaCLOUD Zernike Polynomials are defined as follows:
are the Normalization Coefficients. These can be set to 1 to keep the polynomials Un-Normalized. Or for those who want the Zernike Polynomials to remain normalized, these coefficients are:
are the Radial Zernike Polynomials, which are defined by the following sum:
An important consideration when dealing with these polynomials is that the radius (r) of the polynomial function must be normalized to the unit circle, as Zernike polynomials are orthogonal over the unit-circle. Therefore Zernike Surfaces are regularized to their respective aperture maximum radius. An example of this regularization is shown below, where each shape is shown to fit within the unit circle.

Numerical Validation
We have also verified the numerics for the first 37 Noll Zernike Polynomials and we have the Errors tabulated below with Zernike term normalization and we tabulate the error for 1000 points along a line segment from 0 to Rmax at 20 degrees. This is the Floating point (double precision) error of two methods we calculate the first 37 Zernike functions using extended precision and explicit equations and then compare this result to double precision result using Hypergeometric functions. Please note in Double precision floating point the Significand/Mantissa has 52 bits stored in IEEE 754 which means that relative to 1 in decimal should have about 15 digits, and in special cases this can be up to 17 significant digits.
Below is an image from the second part of the numerical analysis, because in this part of the analysis we find that the radial component is the driver for numerical error. We plot the average error and the absolute error.

Upon fitting this data we get a fit-line with an R^2 of 0.984 for the following Equation:
If we extrapolate out a little bit we can calculate the numerical Sag error for 1nm. Since these calculations are done in mm, the maximum Noll index Zernike can be calculated by solving for idx:
This would correspond to a Noll index of 6.75e10 with a Zernike Coefficient of 1 on Normalized Zernikes.
1
0,0
0
2
1,1
4e-16
3
1,-1
2e-16
4
2,0
1.219e-15
5
2,-2
7.7e-16
6
2,2
8.1e-16
7
3,-1
8.7e-16
8
3,1
2.234e-15
9
3,-3
1.178e-15
10
3,3
5.6e-16
11
4,0
3.499e-15
12
4,2
3.498e-15
13
4,-2
2.570e-15
14
4,4
7.6e-16
15
4,-4
1.716e-15
16
5,1
6.887e-15
17
5,-1
2.582e-15
18
5,3
2.828e-15
19
5,-3
5.797e-15
20
5,5
9.1e-16
21
5,-5
2.612e-15
22
6,0
7.984e-15
23
6,-2
6.514e-15
24
6,2
7.682e-15
25
6,-4
6.967e-15
26
6,4
1.416e-15
27
6,-6
2.450e-15
28
6,6
1.030e-15
29
7,-1
4.737e-15
30
7,1
1.3067e-14
31
7,-3
1.1700e-14
32
7,3
6.531e-15
33
7,-5
8.700e-15
34
7,5
1.908e-15
35
7,-7
2.342e-15
36
7,7
2.262e-15
37
8,0
1.3135e-14
From this data we can see that n seems to be the biggest contributor on loss of precision, therefore we should look explicitly at the radial part of the Zernike polynomial. We perform the same analysis at 0deg angle to get a better understanding of numerical precision as Polynomial order gets quite large:
2
1
7.4340E-16
4
2
1.0199E-15
7
3
1.9477E-15
11
4
2.1583E-15
16
5
3.7206E-15
22
6
3.3561E-15
29
7
6.6609E-15
37
8
6.9594E-15
46
9
1.1183E-14
56
10
1.1904E-14
67
11
1.6403E-14
79
12
1.6981E-14
92
13
2.2165E-14
106
14
2.3049E-14
121
15
2.9538E-14
137
16
3.0361E-14
154
17
3.7170E-14
172
18
3.7687E-14
191
19
4.5026E-14
211
20
4.6228E-14
232
21
5.2889E-14
254
22
5.4489E-14
277
23
6.0251E-14
301
24
6.1179E-14
326
25
6.5899E-14
352
26
6.7084E-14
379
27
7.0519E-14
407
28
7.1839E-14
436
29
7.3628E-14
466
30
7.4845E-14
497
31
7.4841E-14
529
32
7.5952E-14
562
33
7.8031E-14
596
34
7.9430E-14
631
35
8.2194E-14
667
36
8.2758E-14
704
37
1.0280E-13
742
38
1.0438E-13
781
39
9.6065E-14
821
40
9.7141E-14
862
41
1.2747E-13
904
42
1.2943E-13
947
43
1.1515E-13
991
44
1.1606E-13
1036
45
1.5433E-13
References
J. Schwiegerling, Optical specification, fabrication, and testing, SPIE, Bellingham, Washington (1000 20th St. Bellingham WA 98225-6705 USA) (2014).
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