Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.
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ϵ is the obscuration ratio of an annular pupil, b is the aspect ratio of an elliptical pupil, and a is the semiwidth of a rectangular pupil. For a square pupil, .
Table 2
Elliptical Coefficients and the Corresponding Zernike Coefficients when an Elliptical Wavefront with an Aspect Ratio Is Fitted with a Different Number of Polynomials J Showing Independence of the Former but Dependence of the Latter on Ja
j
Elliptical Coefficients
Zernike Coefficients
1
1.0781
1.0781
1.0781
1.5333
1.3853
1.0949
2
0.0501
0.0501
0.0501
0.0504
3
4
1.4447
1.4447
1.4447
1.6658
1.4681
1.0692
5
6
0.1945
0.1945
0.3227
0.7057
7
0.2861
0.2861
1.3946
1.3946
8
9
10
0.1066
0.1066
0.1479
0.1479
11
12
0.3795
0.5000
13
0.3703
0.3325
14
0.1033
0.1668
15
0.2585
0.4086
σ
2.2177
2.7845
2.8481
2.6324
3.1475
3.1647
Also shown is the sigma value for each column as obtained by summing the squares of the coefficient values (excluding the piston coefficient) and taking the square root.
Table 3
Square Coefficients and the Corresponding Zernike Coefficients to Illustrate the Relation between the Piston Coefficient and Wavefront Meana
j
Square Coefficients
Zernike Coefficients
1
1.2328
2
0.2561
3
4
5
6
0.2938
7
2.9374
0.3254
8
9
0.0283
6.5809
10
0.9283
11
1.2938
12
13
1.2937
14
0.2937
15
1.8971
16
17
0.2839
18
19
1.2384
3.5106
20
0.2938
1.2331
21
22
0.1929
23
24
0.0349
0.2195
25
0.9273
2.6426
26
27
28
0.0283
0.1823
σ
5.6895
23.4222
The value of sigma for each column, obtained as in Table 2, is also shown.
Table 4
Annular Coefficients and the Corresponding Zernike Coefficients of an Annular Wavefront with an Obscuration Ratio to Illustrate the Relation between the Coefficients and the Wavefront Variancea
j
Annular Coefficients
Zernike Coefficients
1
0.1828
0.4205
2
0.2347
0.3866
3
0.2374
0.4750
4
5
0.2974
0.1960
6
0.2937
0.4288
7
8
9
0.2947
0.1906
10
0.2937
0.2035
11
0.3074
12
0.2837
13
0.0283
0.0007
14
15
16
0.1273
0.1256
17
0.2274
0.3115
18
19
0.1263
0.0697
20
0.2837
0.1852
21
0.1287
0.0649
22
23
0.0283
24
0.8276
1.0911
25
0.0374
26
27
0.2837
0.1855
28
0.1737
0.0994
σ
1.4709
4.4913
The value of sigma for each column, obtained as in Table 2, is also shown.
Table 5
Hexagonal Coefficients and the Corresponding Zernike Coefficients as Calculated from Eq. (13)a
j
Hexagonal Coefficients
Zernike Coefficients
1
0.0842
0.0329
2
0.0501
3
4
0.0534
5
6
0.1956
0.3974
7
0.0874
0.1100
8
9
10
0.1071
0.1249
11
12
0.2819
0.3293
13
0.0730
0.1216
14
0.1055
0.1599
15
0.0596
0.0903
σ
0.6068
0.7718
The value of sigma for each column, obtained as in Table 2, is also given.
Table 6
Examples of Realistic Aberration Types and the Coefficients of Orthonormal Polynomials
Types
Input Coefficients
Sigma
1
1.0000
2
,
0.9986
3
,
1.0004
Table 7
Hexagonal Coefficients and Zernike Coefficients Calculated by Different Methodsa
ϵ is the obscuration ratio of an annular pupil, b is the aspect ratio of an elliptical pupil, and a is the semiwidth of a rectangular pupil. For a square pupil, .
Table 2
Elliptical Coefficients and the Corresponding Zernike Coefficients when an Elliptical Wavefront with an Aspect Ratio Is Fitted with a Different Number of Polynomials J Showing Independence of the Former but Dependence of the Latter on Ja
j
Elliptical Coefficients
Zernike Coefficients
1
1.0781
1.0781
1.0781
1.5333
1.3853
1.0949
2
0.0501
0.0501
0.0501
0.0504
3
4
1.4447
1.4447
1.4447
1.6658
1.4681
1.0692
5
6
0.1945
0.1945
0.3227
0.7057
7
0.2861
0.2861
1.3946
1.3946
8
9
10
0.1066
0.1066
0.1479
0.1479
11
12
0.3795
0.5000
13
0.3703
0.3325
14
0.1033
0.1668
15
0.2585
0.4086
σ
2.2177
2.7845
2.8481
2.6324
3.1475
3.1647
Also shown is the sigma value for each column as obtained by summing the squares of the coefficient values (excluding the piston coefficient) and taking the square root.
Table 3
Square Coefficients and the Corresponding Zernike Coefficients to Illustrate the Relation between the Piston Coefficient and Wavefront Meana
j
Square Coefficients
Zernike Coefficients
1
1.2328
2
0.2561
3
4
5
6
0.2938
7
2.9374
0.3254
8
9
0.0283
6.5809
10
0.9283
11
1.2938
12
13
1.2937
14
0.2937
15
1.8971
16
17
0.2839
18
19
1.2384
3.5106
20
0.2938
1.2331
21
22
0.1929
23
24
0.0349
0.2195
25
0.9273
2.6426
26
27
28
0.0283
0.1823
σ
5.6895
23.4222
The value of sigma for each column, obtained as in Table 2, is also shown.
Table 4
Annular Coefficients and the Corresponding Zernike Coefficients of an Annular Wavefront with an Obscuration Ratio to Illustrate the Relation between the Coefficients and the Wavefront Variancea
j
Annular Coefficients
Zernike Coefficients
1
0.1828
0.4205
2
0.2347
0.3866
3
0.2374
0.4750
4
5
0.2974
0.1960
6
0.2937
0.4288
7
8
9
0.2947
0.1906
10
0.2937
0.2035
11
0.3074
12
0.2837
13
0.0283
0.0007
14
15
16
0.1273
0.1256
17
0.2274
0.3115
18
19
0.1263
0.0697
20
0.2837
0.1852
21
0.1287
0.0649
22
23
0.0283
24
0.8276
1.0911
25
0.0374
26
27
0.2837
0.1855
28
0.1737
0.0994
σ
1.4709
4.4913
The value of sigma for each column, obtained as in Table 2, is also shown.
Table 5
Hexagonal Coefficients and the Corresponding Zernike Coefficients as Calculated from Eq. (13)a
j
Hexagonal Coefficients
Zernike Coefficients
1
0.0842
0.0329
2
0.0501
3
4
0.0534
5
6
0.1956
0.3974
7
0.0874
0.1100
8
9
10
0.1071
0.1249
11
12
0.2819
0.3293
13
0.0730
0.1216
14
0.1055
0.1599
15
0.0596
0.0903
σ
0.6068
0.7718
The value of sigma for each column, obtained as in Table 2, is also given.
Table 6
Examples of Realistic Aberration Types and the Coefficients of Orthonormal Polynomials
Types
Input Coefficients
Sigma
1
1.0000
2
,
0.9986
3
,
1.0004
Table 7
Hexagonal Coefficients and Zernike Coefficients Calculated by Different Methodsa